Past exam of the physics A course of the University of Cambridge / 2019 / Paper 2 / Section B / B9 / b

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- Solution b

Solution

For the trial displacement

z = sin (q y) e^{i (k x - ω t)}

, direct differentiation gives

\frac{\partial ^{2} z}{\partial x ^{2}} = - k^{2} z, \frac{\partial ^{2} z}{\partial y ^{2}} = - q^{2} z, \frac{\partial ^{2} z}{\partial t ^{2}} = - ω^{2} z .

The two-dimensional wave equation is therefore satisfied if

ω^{2} = \frac{T}{σ} (k^{2} + q^{2}) = c_{m}^{2} (k^{2} + q^{2}), c_{m} = T / σ .

Here

k

is the longitudinal wavenumber,

q

the transverse wavenumber, and

c_{m}

the speed of an unconstrained membrane wave. The physical displacement is the real part, multiplied by an arbitrary displacement amplitude.

T

has units

N m^{- 1}

and

σ

is surface mass density in

kg m^{- 2}

, so

T / σ

has units

m^{2} s^{- 2}

as required.

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