Sound Waves — Question 8

Question 8

A moving particle pattern can remain purely longitudinal

A plane sound wave propagates along $\hat{\mathbf n}=(3\hat{\mathbf x}+4\hat{\mathbf y})/5$. Its particle displacement is $\boldsymbol\xi=a\hat{\mathbf n}\cos(k\hat{\mathbf n}\cdot\mathbf r-\omega t)$ with $a=5.0\,\mu\mathrm m$.

1. Find the Cartesian displacement amplitudes and the displacement perpendicular to propagation.

2. Explain whether a graph of $\xi_y$ versus $x$ would make this a transverse wave. Compute the phase gradient along the $x$ axis.

Question 8 - Solution

A moving particle pattern can remain purely longitudinal

(a) Method and calculation

Project the displacement vector onto coordinate axes:$$a_x=\frac35a=3.0\,\mu\mathrm m,\qquad a_y=\frac45a=4.0\,\mu\mathrm m.$$A perpendicular unit vector is $\hat{\mathbf t}=(-4\hat{\mathbf x}+3\hat{\mathbf y})/5$, giving $\boldsymbol\xi\cdot\hat{\mathbf t}=a(-12+12)\cos\Phi/25=0$.

(b) Interpretation and further calculation

Transverse versus longitudinal is defined relative to propagation, not the plotted coordinate. Here $\boldsymbol\xi\parallel\hat{\mathbf n}$. Along the $x$ axis,$$\Phi=\frac{3k}{5}x-\omega t,\quad \frac{\partial\Phi}{\partial x}=\frac{3k}{5},\quad\lambda_x=\frac{5\lambda}{3}.$$The apparent spatial period along an oblique sampling line exceeds the wavelength measured along the propagation direction.

Physical check

The amplitudes satisfy $a_x^2+a_y^2=a^2$. Rotating the coordinate axes cannot change wave polarization.