Question

Verify that the solution given in equation (7.6) satisfy differential equations (7.5) as well as the required boundary conditions.

Short answer

Both the differential equation and the required boundary conditions are satisfied.

Step-by-step solution

Concept

Write the solution of standing waves in the 1D infinitely.

$$\begin{gathered} F\left(x\right)=A_x\sin \frac{n_x\pi x}{L_x} \\ G\left(y\right)=A_y\sin \frac{n_x\pi y}{L_y} \\ H\left(z\right)=A_z\sin \frac{n_x\pi z}{L_z} \end{gathered}$$

All three equations are the same and differ only in having either x, or y, or as the independent variable. Additionally, all three equations have the same form for the differential equations that each must be proved to fulfill. To prove that the first equation, for F(i) satisfies the appropriate differential equation, is all that is necessary.

$\frac{d^{2}F\left(x\right)}{d{x}^2}=C_{x}F\left(x\right)$

Determine second derivative of the function

Evaluate the second derivative of function F(x),

$$\begin{gathered} \frac{d^{2}F\left(x\right)}{d{x}^2}=\frac{d^2}{d{x}^2}\left(A_x\sin \frac{n_x\pi x}{L_x}\right) \\ =-{\left(\frac{n_x\pi }{L_x}\right)}^2\left(A_x\sin \frac{n_x\pi x}{L_x}\right) \\ =-{\left(\frac{n_x\pi }{L_x}\right)}^{2}F\left(x\right) \end{gathered}$$

The above equation is same as the required differential equation.

$\frac{d^{2}F\left(x\right)}{d{x}^2}=C_{x}F\left(x\right)$

It provide that,

$C_x=-{\left(\frac{n_x\mathrm{\pi }}{L_x}\right)}^2$

The boundary conditions to be satisfied are that $F\left(x\right)=0$at $x=0$and at $x=L_x$.

Determine F (x)at x = 0.

role="math" localid="1659781703280" $$\begin{gathered} F\left(0\right)=A_x\sin 0 \\ =0 \end{gathered}$$

Similarly, at x = L ,

$$\begin{gathered} F\left(L_x\right)=A_x\sin \frac{n_x\pi L_x}{L_x} \\ =A_x\sin n_x\pi \end{gathered}$$

Which is zero because$n_x\pi$ is an integer.

Therefore, both the differential equation and the required boundary conditions are satisfied.