Prismatic horn
Contents
In a prismatic horn the cross-sectional dimensions are constant in the principal direction of vibration.
The following discussion is for a longitudinally resonant thin (slender) prismatic horn. (A thin horn is one whose transverse dimensions (e.g., diameter) are sufficiently small at a specified frequency that any energy from transverse motion due to Poisson coupling can be neglected.)
Amplitude and stress
In such a horn the amplitude is cosinually distributed along the horn's length from the free end —
\begin{align} \label{eq:10701a} u(x) &= U_p \cos \left( {\frac{2\pi}{\lambda_{tw}}} \, x \right) \end{align}
where —
| \( x \) | = Distance, starting from a free end (antinode) |
| \( u \) | = Amplitude distribution |
| \( U_p \) | = Maximum (peak) amplitude |
| \( \lambda_{tw} \) | = Thin-wire wavelength at the resonant frequency |
The quantity in ( ) is in radians. The amplitude \( u(x) \) is maximum at the antinodes (the free end and every half wavelength (\( \lambda_{tw}/2 \)) thereafter). \( u(x) \) is zero at the nodes (\( \lambda_{tw}/4 \) and every \( \lambda_{tw}/2 \) thereafter).
The stress is the product of Young's modulus \( E \) and the strain \( \epsilon \), where the strain is the derivative (slope) of the amplitude curve —
\begin{align} \label{eq:10702a} {\sigma}(x) &= E \, \epsilon \\[0.7em]%eqn_interline_spacing &= E \, \frac{du}{dx} \nonumber \\[0.7em]%eqn_interline_spacing &= E \, \left[ U_p \left( {\frac{2\pi}{\lambda_{tw}}} \right) \sin \left( {\frac{2\pi}{\lambda_{tw}}} \, x \right) \right] \nonumber \end{align}
The stress \( {\sigma}(x) \) is zero at the antinodes (the free end and every half wavelength (\( \lambda_{tw}/2 \)) thereafter). \( {\sigma}(x) \) is maximum at the nodes (\( \lambda_{tw}/4 \) and every \( \lambda_{tw}/2 \) thereafter).
\begin{align} \label{eq:10707a} {\sigma}_{max} &= E \, \left[ U_p \left( {\frac{2\pi}{\lambda_{tw}}} \right) \right] \end{align}
Rather than expressing the above equations in terms of the wavelength, they may be more conveniently expressed in terms of the frequency and wave speed where it is recognized that —
\begin{align} \label{eq:10703a} \lambda_{tw} = \frac{c_{tw}}{f} \end{align}
where —
| \( c_{tw} \) | = Thin-wire wave speed = \( \sqrt{\frac{E}{\rho}} \) |
| \( f \) | = frequency |
Then equations \eqref{eq:10701a} through \eqref{eq:10707a} become —
\begin{align} \label{eq:10704a} u(x) &= U_p \cos \left( {\frac{2\pi \, f}{c_{tw}}} \, x \right) \end{align}
\begin{align} \label{eq:10705a} {\sigma}(x) &= E \, \left[ U_p \left( {\frac{2\pi \, f}{c_{tw}}} \right) \sin \left( {\frac{2\pi \, f}{c_{tw}}} \, x \right) \right] \end{align}
\begin{align} \label{eq:10706a} {\sigma}_{max} &= E \, \left[ U_p \left( {\frac{2\pi \, f}{c_{tw}}} \right) \right] \end{align}
Tuning
The tuning of a prismatic resonator depends on the ratio of the resonator's diameter to its prismatic half wavelength.
Tuned length
The tuned length of a thin prismatic resonator is —
\begin{align} \label{eq:10710a} \Gamma_{tw} = \frac{c_{tw}}{2 \, f} \end{align}
where —
| \( \Gamma_{tw} \) | = thin-wire half-wavelength |
| \( c_{tw} \) | = thin-wire wave speed |
| \( f \) | = frequency |
The thin-wire wave speed can be determined from the material properties —
\begin{align} \label{eq:10711a} c_{tw} &= \sqrt{\frac{E}{\rho}} \end{align}
where —
| \( E \) | = modulus of elasticity (Young's modulus) |
| \( \rho \) | = density |
As the lateral dimensions of the resonator increase (i.e., the resonator is no longer "thin"), the half-wavelength decreases below that of a thin wire. From a physical standpoint, when the resonator's diameter is larger than an infinitely thin wire then radial "breathing" occurs due to Poisson's coupling. This radial motion imparts additional kinetic energy which would cause the resonator's frequency to drop. In order to maintain the desired frequency the resonator's length must be reduced.
At larger diameters even more breathing occurs and the resonator must tune progressively shorter. At the largest possible diameter the resonator is reduced to a thin flat disk. Then the breathing is simply the fundamental radial resonance of the disk.
The actual tuned length at a finite diameter \( D \) can be determined by multiplying \( \Gamma_{tw} \) by a length factor \( K_L \).
\begin{align} \label{eq:10712a} \Gamma = K_L \,\, \Gamma_{tw} \end{align}
where —
| \( \Gamma \) | = half-wavelength at diameter \( D \) |
| \( \Gamma_{tw} \) | = thin-wire half wavelength |
| \( K_L \) | = tuned length factor |
The tuned length factor can be determined from figures 1 or 2 or the associated curve fit equations. Note that figure 2 is simply a zoomed portion of figure 1 where the horizontal axis is limited to 1.1. This is typically the largest value for cylinders that are longitudinally resonant. The images in the figures show the relative sizes of the cylinders and their axial amplitudes (parallel to the cylinder's axis.)
(Note: Two theoretical approximate length factors — the Rayleigh factor \( K_{L_R} \) and the Mori factor \( K_{L_M} \) — are discussed in Appendix A.)
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Curve fits
In order to fit the FEA data reasonably well, two polynomial curve fits are for the data of figures 1 and 2 —
For \( X \le 1.3 \) —
\begin{align} \label{eq:10713a} K_L = 1 - 0.0255 \, X^2 - 0.2861 \, X^3 + 0.4793 \, X^4 - 0.2932 \, X^5 \end{align}
For \( X \ge 1.3 \) —
\begin{align} \label{eq:10713b} K_L = -451.5 + 670.0 \, X - 248.3 \, X^2 \end{align}
where —
| \( X \) | = Cylinder diameter / Thin-wire half wavelength |
| = \( D/\Gamma_{tw} \) | |
| \( K_L \) | = Tuned length factor |
Example 1
Given — a 20 kHz \( \phi \)90 mm cylindrical resonator. The resonator material has a thin-wire wave speed \( c_{tw} \) of 5100 m/sec and a Poisson's ratio of 0.33.
Determine — the tuned half wavelength \( \Gamma \).
- \( \Gamma_{tw} \) = 127.5 mm (equation \eqref{eq:10710a})
- \( D/\Gamma_{tw} \) = 90.0/127.5 = 0.706
- \( K_L \) = 0.954 (from either figure 1 and 2 (which are appropriate for this material) or from equation \eqref{eq:10713a})
- \( \Gamma \) = 0.954 * 127.5 mm = 121.6 mm (from equation \eqref{eq:10712a})
Tuning rate
From Appendix B the tuning rate for a thin prismatic resonator is —
\begin{align} \label{eq:10715a} \varphi_{tw} = -\frac{f}{L} \end{align}
where —
| \( \varphi_{tw} \) | = thin-wire tuning rate [Hz/mm] |
| \( f \) | = current frequency [Hz] |
| \( L \) | = current length [mm] |
Note that the tuning rate is always negative as indicated by the minus sign. However, the tuning rate is generally discussed as a positive (absolute) value.
As the lateral dimensions of the resonator increase (i.e., the resonator is no longer "thin"), the tuning rate decreases (i.e., more material must be removed in order to achieve a given frequency change). This can be accommodated by applying a tuning rate factor to equation \eqref{eq:10715a} —
\begin{align} \label{eq:10716a} \varphi &= K_T \, \varphi_{tw} \\[0.3em]%eqn_interline_spacing &= K_T \, \left( \frac{f}{L} \right) \nonumber \end{align}
Note that equations of this section only apply if the horn is prismatic. They do not apply if the horn has gain.
The tuning rate factor can be taken from figures 3 and 4 or the associated curve fit equation.
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Curve fit
The polynomial curve fit for the data of figures 3 and 4 is —
\begin{align} \label{eq:10717a} K_T = 1 - 0.0323 \, X^2 - 0.0987 \, X^3 + 0.0262 \, X^4 - 0.1377 \, X^5 \end{align}
where —
| \( X \) | = Cylinder diameter / Thin-wire half wavelength |
| = \( D/\Gamma_{tw} \) | |
| \( K_T \) | = Tuning rate factor |
Example 2
Given — a \( \phi \)90 mm x half-wave cylindrical resonator. The length is 124.8 mm for which the frequency is 19500 Hz. The resonator material has a thin-wire wave speed \( c_{tw} \) of 5100 m/sec and a Poisson's ratio of 0.33.
Determine — the tuning rate \( \varphi \) and expected final length.
- \( \Gamma_{tw} \) = 130.8 mm (equation \eqref{eq:10710a})
- \( D/\Gamma_{tw} \) = 90.0 / 130.8 = 0.688
- \( K_T \) = 0.94 (from either figure 3 and 4 (which are appropriate for this material) or from equation \eqref{eq:10717a})
- \( f/L \) = 19500 Hz / 124.8 mm = 156 Hz/mm (thin-wire tuning rate)
- \( \varphi \) = 0.94 * 156 Hz/mm = 147 Hz/mm (from equation \eqref{eq:10716a})
Appendix A. Tuned length of a cylindrical prismatic resonator
(Rayleigh and Mori approximations)
For a thin wire the half-wavelength for a pressure wave is —
\begin{align} \label{eq:10750a} \Gamma_{tw} = \frac{c_{tw}}{2 \, f} \end{align}
where —
| \( \Gamma_{tw} \) | = thin-wire half-wavelength |
| \( c_{tw} \) | = thin-wire wave speed |
| \( f \) | = frequency |
The thin-wire wave speed can be determined from the material properties —
\begin{align} \label{eq:10751a} c_{tw} &= \sqrt{\frac{E}{\rho}} \end{align}
where —
| \( E \) | = modulus of elasticity (Young's modulus) |
| \( \rho \) | = density |
As the lateral dimensions of the resonator increase (i.e., the resonator is no longer "thin"), the half-wavelength decreases below that of a thin wire. From a physical standpoint, when the resonator's diameter is larger than an infinitely thin wire then radial "breathing" occurs due to Poisson's coupling. This imparts additional kinetic energy which would cause the resonator's frequency to drop. In order to maintain the desired frequency the resonator's length must be reduced.
At larger diameters even more breathing occurs and the resonator must tune progressively shorter. At the largest possible diameter the resonator is reduced to a thin flat disk. Then the breathing is simply the fundamental radial resonance of the disk.
Rayleigh approximation
Rayleigh (1) (vol. 1, pp. 251 - 252) first developed an equation that approximately accounts for this phenomena for a prismatic cylinder.
\begin{align} \label{eq:10752a} \Gamma = K_{L_R} \,\, \Gamma_{tw} \end{align}
where —
| \( \Gamma \) | = half-wavelength at cylinder diameter \( D \) |
| \( \Gamma_{tw} \) | = thin-wire half wavelength |
| \( K_{L_R} \) | = Rayleigh length factor |
\begin{align} \label{eq:10753a} K_{L_R} = 1 - \small\frac{1}{8} \normalsize \left(\pi \, \nu \, \frac{D}{\Gamma_{tw}} \right)^2 \end{align}
where —
| \( D \) | = resonator diameter |
| \( \nu \) | = Poisson's ratio |
Note that \( K_{L_R} \) depends on the ratio of the resonator's diameter (\( D \)) to the thin-wire half wavelength (\( \Gamma_{tw} \)). When \( D = 0 \), \( K_{L_R} = 1 \) so that \( \Gamma = \Gamma_{tw}\) (i.e., an exact solution). However, as the resonator diameter increases the Rayleigh approximation first under-estimates and then over-estimates the tuned length (figures A1 and A2 for a typical acoustic material).
Mori approximation
Mori (1) improved on Rayleigh's effort by developing an equation that is correct both for a thin wire and, at large diameters, for a radially vibrating disk. The results at intermediate diameters are approximated by a method called "apparent elasticity".
\begin{align} \label{eq:10754a} \Gamma = K_{L_M} \,\, \Gamma_{tw} \end{align}
where —
| \( \Gamma \) | = half-wavelength at cylinder diameter \( D \) |
| \( K_{L_M} \) | = Mori length factor |
\begin{align} \label{eq:10755a} K_{L_M} = \left[ \frac{1 - B_1 \, \psi}{1 - B_2 \, \psi} \right]^{1/2} \end{align}
where —
| \( B_1 \) | = \( 1 - \nu^2 \) |
| \( B_2 \) | = \( 1 - 3 \, \nu^2 - 2 \, \nu^3 \) |
| \( \nu \) | = Poisson's ratio |
| \( \psi \) | \begin{align} \label{eq:10756a} &= \left[ \frac{ \large\frac{\pi}{2} \left(\large\frac{D}{\Gamma_{tw}}\right)}{\alpha} \right]^{2} \nonumber \\[0.7em]%eqn_interline_spacing \end{align} |
| \( D \) | = resonator diameter |
| \( \alpha \) | \( \approx 1.84 + 0.68 \, \nu \) [Derks, p. 42, eqn. 5.8] |
| \( \alpha \) | \( \approx 1.85 \left(1 + 0.386 \, \nu - 0.146 \, \nu^2 + 0.115 \, \nu^3 \right) \) [Gladwell (1), p. 345] |
\( K_{L_M} \) is shown in figures A1 and A2 for a typical acoustic material. Note that the Mori approximation consistently under-estimates the tuned length.
Note that \( \alpha \) is approximate. Derks and Gladwell give somewhat different equations. Derks takes his linear equation from a graph by Kleesattel (2), p. 3, figure 1, curve 1. Kleesattel's graph of this curve is not quite linear so Gladwell's approximation may be somewhat better. However, the difference is small — for \( \nu \) = 0.33 (approximately typical for many acoustic materials), Gladwell's equation gives 2.0666 whereas Dirks' equation gives 2.0639. Figures A1 and A2 shows that the results are nearly indistinguishable.
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Appendix B. Tuning rate of a thin prismatic resonator
For a thin prismatic resonator the fundamental wave equation is —
\begin{align} \label{eq:10760a} c_{tw} &= 2 \, \Gamma_{tw} \, f \end{align}
where —
| \( c_{tw} \) | = thin-wire wave speed |
| \( \Gamma_{tw} \) | = thin-wire half wavelength |
| \( f \) | = frequency of longitudinal wave |
Solving for f —
\begin{align} \label{eq:10762a} f &= \frac{c_{tw}}{2 \, \Gamma_{tw} } \\[0.7em]%eqn_interline_spacing &= \small\frac{1}{2} \normalsize \, c_{tw} \, \Gamma_{tw}^{-1} \nonumber \end{align}
Differentiating \( f \) with respect to \( \Gamma_{tw} \) to get the thin-wire tuning rate \( \varphi_{tw} \) —
\begin{align} \label{eq:10763a} \frac{df}{\Gamma_{tw}} = \varphi_{tw} &= -\small\frac{1}{2} \normalsize \, c_{tw} \, \Gamma_{tw}^{-2} \\[0.7em]%eqn_interline_spacing &= -\small\frac{1}{2} \normalsize \, \frac{c_{tw}}{\Gamma_{tw}^2} \nonumber \end{align}
Substituting \( c_{tw} \) from equation \eqref{eq:10760a} into equation \eqref{eq:10763a} —\begin{align} \label{eq:10764a} \varphi_{tw} &= -\frac{f}{\Gamma_{tw}} \end{align}
If the resonator is \( n \) half waves long then the tuning rate is reduced by \( 1/n \) since each tuning slice is effectively divided among each of the \( n \) half-waves —
\begin{align} \label{eq:10765a} \varphi_{tw} &= -\frac{f}{n \, \Gamma_{tw}} \end{align}
Appendix C. Energy stored in a thin longitudinally resonant member
For a longitudinally resonant member whose cross-sectional dimensions are small compared to its half wavelength, the kinetic energy stored in any thin slice \( dx \) is —
\begin{align} \label{eq:10721a} dW &= \small\frac{1}{2} \normalsize \, \dot{u}^2 \, dm \\[0.7em]%eqn_interline_spacing &= \small\frac{1}{2} \normalsize \, \dot{u}^2 \, \left( \rho \, A \, dx \right) \nonumber \end{align}
where —
| \( dW \) | = stored energy in thin slice |
| \( \dot{u} \) | = velocity of slice |
| \( dm \) | = mass of slice |
| \( \rho \) | = density |
| \( A \) | = area of slice |
| \( dx \) | = thickness of slice |
Quarter wave prismatic
For a quarter wave (\( \lambda/4 \)) the kinetic energy can be found by integrating equation \eqref{eq:10721a} over the quarter wave —
\begin{align} \label{eq:10722a} W_{\lambda/4} &= \small\frac{1}{2} \normalsize \, \int_{0}^{\lambda/4}{\dot{u}^2 \, \left( \rho \, A \, dx \right) } \end{align}
For a thin prismatic resonator the amplitude distribution (from the antinodal free end) is —
\begin{align} \label{eq:10723a} u &= U_p \cos \left( {\frac{2\pi}{\lambda}} \, x \right) \, \sin \left( 2\pi f \, t \right) \end{align}
The velocity is the rate of change of the amplitude. Thus, differentiating equation \eqref{eq:10723a} with respect to time gives the velocity distribution —
\begin{align} \label{eq:10724a} \dot{u} &= \frac{du}{dt} \\ &= \left( 2\pi f \right) U_p \, \cos \left( {\frac{2\pi}{\lambda}} \, x \right) \, \cos \left( 2\pi f \, t \right) \nonumber \\ &= \dot{U}_p \, \cos \left( {\frac{2\pi}{\lambda}} \, x \right) \, \cos \left( 2\pi f \, t \right) \nonumber \end{align}
Substituting equation \eqref{eq:10724a} into equation \eqref{eq:10722a} and ignoring the time varying component (to get the maximum energy during the cycle) —
\begin{align} \label{eq:10725a} W_{\lambda/4} &= \small\frac{1}{2} \normalsize \, \int_{0}^{\lambda/4}{\left[ \dot{U}_p \cos \left( {\frac{2\pi}{\lambda}} \, x \right) \right]^2 \, \left( \rho \, A \, dx \right) } \end{align}
For a prismatic resonator the area \( A \) is constant. Also, neither the peak amplitude \( U_p \) nor the frequency \( f \) depend on \( x \). Assume that the density \( \rho \) does not vary with \( x \). Then all of these can be taken outside the integral —
\begin{align} \label{eq:10726a} W_{\lambda/4} &= \small\frac{1}{2} \normalsize \left( \rho \, A\right)\, \dot{U_p}^2 \int_{0}^{\lambda/4}{\left[ \cos \left( {\frac{2\pi}{\lambda}} \, x \right) \right]^2 \, dx } \end{align}
Integrating yields —
\begin{align} \label{eq:10727a} W_{\lambda/4} &= \small\frac{1}{2} \normalsize \left( \rho \, A\right)\, \dot{U_p}^2 {\left[ \frac{x}{2} + \small\frac{1}{4} \normalsize \, \sin \left( {\frac{4\pi }{\lambda}} \, x \right) \right]}_0 ^{\lambda/4} \end{align}
When equation \eqref{eq:10727a} is evaluated for \( x \) between the limits of 0 and \( {\lambda}/4 \), the sin term is 0 at both limits and so drops out. Then equation \eqref{eq:10727a} is —
\begin{align} \label{eq:10728a} W_{\lambda/4} &= \small\frac{1}{2} \normalsize \left( \rho \, A\right)\, \dot{U_p}^2 {\left[ \frac{x}{2} \right]}_0 ^{\lambda/4} \\[0.7em]%eqn_interline_spacing &= \small\frac{1}{2} \normalsize \left( \rho \, A \right)\, \dot{U_p}^2 {\left[ \frac{\lambda}{8} \right]} \nonumber \\[0.7em]%eqn_interline_spacing &= \small\frac{1}{2} \normalsize \left[ \small\frac{1}{2} \normalsize \left( \rho \, A \, \frac{\lambda}{4} \right) \right] \, \dot{U_p}^2 \nonumber \end{align}
The factor in ( ) is just the total mass of the quarter wave section (i.e., \( m_{\lambda/4} \)) so equation \eqref{eq:10728a} can be written as —
\begin{align} \label{eq:10729a} W_{\lambda/4} &= \small\frac{1}{2} \normalsize \left( \frac{m_{\lambda/4}}{2} \right) \, \dot{U_p}^2 \\[0.7em]%eqn_interline_spacing &= \small\frac{1}{2} \normalsize \left( \frac{m_{\lambda/4}}{2} \right) \, \big[ \left( 2\pi f \right) U_p \big]^2 \nonumber \end{align}
Thus, the energy that is cosinually distributed in a prismatic member is the same as if 1/2 of the total quarter-wave mass had been concentrated at the antinode and vibrating with peak velocity \( \dot{U}_p \), corresponding to peak amplitude \( U_p \).
Note that equation \eqref{eq:10729a} could have also been derived from the strain (potential) energy distribution, since at resonance the maximum strain energy equals the maximum kinetic energy.
Half wave prismatic resonator
A half wave (\( \lambda/2 \)) prismatic has twice the mass of a quarter wave prismatic. Therefore it has twice the stored energy of equation \eqref{eq:10729a}—
\begin{align} \label{eq:10730a} W_{\lambda/2} &= \left( \frac{m_{\lambda/2}}{2} \right) \, \dot{U_p}^2 \\[0.7em]%eqn_interline_spacing &= \left( \frac{m_{\lambda/2}}{2} \right) \, \left[ \left( 2\pi f \right) U_p \right]^2 \nonumber \end{align}
Half wave stepped resonator
A half wave resonator with a sharp step exactly at the node can be considered simply as two joined quarter wave resonators — i.e., a quarter wave input section joined to a quarter wave output section. Then from equation \eqref{eq:10729a}—
\begin{align} \label{eq:10731a} W_{\lambda/2}|_{step} &= W_{\lambda/4}|_i + W_{\lambda/4}|_o \\[0.7em]%eqn_interline_spacing &= \left[ \small\frac{1}{2} \normalsize \left( \frac{m_{\lambda/4}|_i}{2} \right) \, \dot{U}|_i^2 \right] + \left[ \small\frac{1}{2} \normalsize \left( \frac{m_{\lambda/4}|_o}{2} \right) \, \dot{U}|_o^2 \right] \nonumber \\[0.7em]%eqn_interline_spacing &= \small\frac{1}{2} \normalsize \left\{ \left[ \left( \frac{m_{\lambda/4}|_i}{m_{\lambda/4}|_o} \right) \, \left( {\frac{\dot{U}|_i}{\dot{U}|_o}} \right)^2 \right] + 1 \right\} \left( \frac{m_{\lambda/4}|_o}{2} \right) \, \dot{U}|_o^2 \nonumber \\[0.7em]%eqn_interline_spacing \end{align}
where —
| \( i \) | = input quarter wave |
| \( o \) | = output quarter wave |
For a resonator with a sharp step at the node, the ratio of input mass to output mass is inversely proportional to the ratio of the of the corresponding velocities —
\begin{align} \label{eq:10732a} \left( \frac{m_{\lambda/4}|_i}{m_{\lambda/4}|_o} \right) &= \left( {\frac{\dot{U}|_o}{\dot{U}|_i}} \right) = G_{stepped} \end{align}
where —
| \( G_{stepped} \) | = theoretical gain of a resonator with a sharp step at the node |
Substituting equation \eqref{eq:10732a} into equation \eqref{eq:10731a} —
\begin{align} \label{eq:10733a} W_{\lambda/2}|_{stepped} &= \small\frac{1}{2} \normalsize \left( \frac{1}{G_{stepped}} + 1 \right) \left( \frac{m_{\lambda/4}|_o}{2} \right) \, \dot{U}|_o^2 \end{align}
Comparison
Comparing the half-wave stepped resonator of equation \eqref{eq:10733a} to the half-wave prismatic resonator of equation \eqref{eq:10730a} where both are specified to have the same output mass (i.e., the same output cross-sectional area) and the same output velocity —
\begin{align} \label{eq:10734a} \frac{W_{\lambda/2}|_{stepped}}{W_{\lambda/2}|_{prismatic}} = \small\frac{1}{2} \left( \frac{1}{G_{stepped}} + 1 \right) \end{align}
Note that when the step disappears (for which the gain of the stepped resonator is just 1.0) the stored energy of the stepped resonator (which is then no longer stepped) is just equal to the stored energy of the prismatic resonator, as required.
When the resonator has gain greater than 1.0 then equation \eqref{eq:10734a} shows that the stored energy of the stepped resonator is less than the stored energy of the prismatic resonator (assuming that both have the same output amplitude). This is dispite the fact that the input section of the stepped resonator must be larger than that of the prismatic resonator in order to provide gain (assuming, again, that both have the same output mass and velocities). It might initially seem odd that a stepped resonator whose input section is more massive than the prismatic resonator would have lower stored energy. However, this occurs because the kinetic energy depends directly on the mass but, in addition, on the square of the velocity. Thus, although the input section is more massive its velocity is proportionately lower (i.e., to provide the required gain) by which the net kinetic energy is lower. For example, ...