Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 1 / 38B (Waves)

Linearized mass conservation, momentum balance and the isentropic pressure-density relation are ${\rho }_t^{\prime }+{\rho }_0\nabla \cdot \mathbf{u}=0$, ${\rho }_0{\mathbf{u}}_t=-\nabla \tilde{p}$, and $\tilde{p}=c_0^2{\rho }^{\prime }$. Differentiate mass conservation in time and take the divergence of momentum balance to eliminate velocity. This gives the acoustic pressure wave equationFor the convention $\tilde{p}=\mathrm{Re}{e}^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)}$, momentum balance gives the complex velocity amplitude $\mathbf{k}/({\rho }_0\omega )$, with $\omega =c_0|\mathbf{k}|$. Thus the incident velocity field is $\mathrm{Re}[{\mathbf{k}}_I{e}^{i({\mathbf{k}}_I\cdot \mathbf{r}-\omega t)}/({\rho }_1\omega )]$ in the lower gas.

Let $a=k_I\sin \theta$, $l_1=k_I\cos \theta >0$, $\omega =c_1{k}_I$, and $l_2=\sqrt{(\omega /c_2{)}^2-a^2}$ with nonnegative real or imaginary part. This chooses an outgoing propagating wave or an evanescent wave decaying as $y\to +\infty$. Conservation of tangential wavevector is the acoustic Snell law for elastic and acoustic waves. Write the complex pressures asInviscid kinematic matching requires equal normal velocities at the plate, not equal tangential velocities:Define the normal acoustic impedance ratio $\delta ={\rho }_2{l}_1/({\rho }_1{l}_2)$, so $T=\delta (1-R)$. For a propagating transmitted wave this is ${\rho }_2{c}_2\cos \theta /({\rho }_1{c}_1\cos {\theta }_2)$. The dynamic condition giveswhich is exactly the printed dimensionless parameter. Solving yields the pressure reflection coefficient and transmission coefficientAt $l_2=0$ use the preceding velocity equations, or take their limit, rather than substituting an infinite $\delta$. For real $l_2>0$, the reflected energy fraction is $|R{|}^2$, the transmitted fraction is $|T{|}^2/\delta$, and these sum to one. An evanescent transmitted wave carries no mean normal energy flux.

When the gases match, $\delta =1$, so $R=-i\beta /(2-i\beta )$ and $T=2/(2-i\beta )$. For a nongrazing incident wave of nonzero frequency, perfect transmission means $R=0$, equivalent toThe plate's inertial and bending terms cancel: the incident frequency and tangential wavevector coincide with its free flexural wave dispersion relation. Then $T=1$, including its phase. With positive $m,B$, this condition is possible only at angles and frequencies satisfying that relation; for normal incidence at nonzero frequency it cannot occur unless the mass is zero.

Away from a transmitted grazing or critical angle, ${\rho }_1{c}_1\gg {\rho }_2{c}_2$ gives a small normal acoustic impedance ratio $\delta$. ThusThe reflected energy is nearly all the incident energy, while the pressure phase depends on the plate response. Large $|\beta |$ gives $R\simeq +1$, resembling a rigid boundary; $\beta \simeq 0$ gives $R\simeq -1$, resembling a pressure-release boundary. Bulk impedance contrast alone does not ensure $\delta \ll 1$ arbitrarily close to a transmitted critical angle, since the factor $1/\cos {\theta }_2$ can be large.

For fixed frequency and finite plate parameters, $\beta$ contains the incident normal wavevector and therefore tends to zero as $\theta \to \pi /2$. If $c_1\ne c_2$, the outgoing $l_2$ tends to a nonzero real or imaginary value, whereas $l_1\to 0$, so $\delta \to 0$ too. ConsequentlyThe lower-side boundary pressure $1+R$ tends to zero, regardless of how large the fixed $m,B$ are. The angle must become closer to grazing as those parameters grow; the assertion is not uniform in an unbounded plate mass or rigidity.

If $c_1=c_2$, however, $l_2=l_1$ and $\delta ={\rho }_2/{\rho }_1$ remains constant. The exact grazing limit is $(\delta -1)/(\delta +1)$, rather than exactly $-1$ for finite contrast. Under the stated large-contrast approximation it is still arbitrarily close to the pressure-release value, with boundary pressure $2\delta /(1+\delta )$. Thus the printed pressure-release description is an approximation in that equal-speed case, not an exact zero-pressure identity. If a transmitted wave becomes evanescent, the outgoing square-root convention above continues to give the unequal-speed grazing limit.