Solved Problems IV (Wave Phenomenon)

1.      General expression for disturbance of a harmonic vibratory rope with angular frequency ω and circular wave-number $k={\omega }/{c}\;$ is given by: $g(x,t)=A\cos \{(kx-\omega t)+{{\phi }_{0}}\}+B\cos \{(kx+\omega t)+{{\phi }_{1}}\}$, the so-called standing waves.

The boundary conditions in this case, wherein the rope is driven at x = 0 and fixed at x = L, are:

At x = 0: $g(0,t)={{A}_{0}}\cos \omega t$

At x = L: $g(L,t)=0$

The complex representation of the disturbance on the rope is: $G(x,\omega )=C{{e}^{+ikx}}+D{{e}^{-ikx}}$, here, C and D are complex constants.

a.       Give C and D in terms of real constants: A, B, ${{\phi }_{0}}$, and ${{\phi }_{1}}$.

b.      Give the boundary conditions for G(x,ω) at x = 0 and at x = L. Define C and D in this case.

c.       ‘Eigen-modes’ of the rope are possible for a defined angular-frequencies ω_r. For these frequencies, with the corresponding values of $k={{{\omega }_{r}}}/{c}\;$, as A₀ is moving to 0 a finite solution of g(x,t) is exist (except at x = 0 and at x = L). Define the value of k and ω_r for possible eigen-modes with the help of the boundary conditions: $G(0,\omega )=G(L,\omega )=0$.

d.      Define for ω ≠ ω_r with the values of C and D from (b), the complex expression for disturbance G(x,ω) in terms of A₀, k and L.

e.       Define g(x,t).

Solution:

 

1.      A rectangular membrane is fixed on its four sides. The coordinates of its corner are: (0,0); (a,0); (a,b); and (0,b). The disturbance g(x,y) on the membrane satisfies the wave equation:

$\frac{{{\partial }^{2}}g}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}g}{\partial {{y}^{2}}}=\frac{1}{{{c}^{2}}}\frac{{{\partial }^{2}}g}{\partial {{t}^{2}}}$

On its edge applies that g = 0. We assume that the membrane vibrates harmonically with angular frequency ω.

a.    The complex amplitude G(x,y) satisfies a differential equation. Please derive this differential equation.

b.      Assume that the complex amplitude G(x,y) can be written as a product of:

$G(x,y)={{G}_{0}}{{H}_{1}}(x){{H}_{2}}(y)={{G}_{0}}({{e}^{i{{k}_{x}}x}}+p{{e}^{-i{{k}_{x}}x}})({{e}^{i{{k}_{y}}y}}+q{{e}^{-i{{k}_{y}}y}})$

i.                Define the boundary conditions for H₁(x) and H₂(y).

ii.              Define from (i) first the conditions for p and q, then show that for k_x and k_y the conditions: ${{k}_{x}}=\frac{\pi m}{a}$ and ${{k}_{y}}=\frac{\pi n}{b}$ with m = 0,1,2,3,... and n = 0,1,2,3,... are valid.

c.    For which angular frequency ω = ω_nm in the wave equation is valid? Also, give the lowest frequency f₀ whereby the wave equation is valid.

d.    Define the disturbance g(x,y,t) as a function of the coordinates when the membrane vibrates harmonically with the lowest frequency f₀ and the maximum disturbance at the center of the membrane is g₀.

Solution: