Solving More Difficult Partial Differential Equations: Plane Waves & The Rectangular Membrane (Pt. 5, Section i)

 We examine now the case of a rectangular frame which is capable of sustaining plane vibrations, for which we seek to obtain relevant solutions.  The frame is shown below:

                                                                           

Where a is the x dimension and b the y-dimension. We have a relevant wave function such that:  u  =  u (x, y, t)

So can be written as the partial differential equation:

¶ ² u/ ¶ t ²  =   c ²   [¶ ² u/ ¶ x² -   ¶ ² u/ ¶ y²    ]

And we require:  

u(0, y, t) =  0   (t > 0)

and: u(x, 0, t) = 0 (t > 0)

We write in variables separable (condensed) form:

 u(x, y, t) =    X(x)  Y(y)  T(t)

Whence:

¶ ² u/ ¶ t ²  =  XY T'', 

 ¶ ² u/ ¶ x² = X'' YT,      

¶ ² u/ ¶ y² =  XY''T

Þ   

XY T'' =  c ²   (X'' YT  +    XY''T)  

Now, separate variables to get:

X''/X =  1/ c ²  (T''/ T  -   Y''/ Y)  =  g  =   const.  =   - b²

Þ   

X''  +    b² X =  0

Then:

u(0, y, t) =  X(0) Y(y) T(t) = 0

  u(a, y, t) =  X(a) Y(y) T(t) = 0

For the other eqn.:

Y''/ Y  =  1/ c ²  (T''/ T )   +  b²  =  const.  =   - a²

From which we obtain:

i) Y''  +    a²Y   = 0

ii) T'' +  c ²   (a²   +   b² ) T = 0

In effect we now have   separate equations corresponding to the 3 variables. And for the solutions we will need the respective forms:

X(x)  =  A cos   bx +   B sin  b x

Y(y)  =  C cos  a y  +  D sin   ay

T(t)  =  E cos gt    +  F sin   g t

Where:   g² =  c ²  (a²   +   b² )

Problem:

At this point, use the equation for u(0, y, t) to obtain the value of A, and thence find X(x).

We will continue Sec. ii based on these results!