Engineering Acoustics/Acoustic wave solution in Cartesian coordinates

The one dimensional acoustic wave equation is described by the following second order partial differential equation.

$\frac{{\partial }^{2}P}{\partial x^2}=\frac{1}{C_o^2}\frac{{\partial }^{2}P}{\partial t^2}$

It can be solved using separation of variables also known as the Fourier method. Suppose the pressure is the product of one function only dependent on space and another function only dependent on time.

$P(x,t)=X(x)T(t)$

Substituting back into the wave equation

$X^{″}T=\frac{1}{C_o^2}X{T}^{″}$

$\frac{X^{″}}{X}=\frac{1}{C_o^2}\frac{T^{″}}{T}=-{\lambda }^2$

This substitution leads to two homogeneous second order ordinary differential equations, one in time and one in space.

$T^{″}+C_o^2{\lambda }^{2}T=0$

$X^{″}+{\lambda }^{2}X=0$

The time function is expected to be dependent on the angular frequency of the wave.

$T=C{e}^{j\omega t}$

Substituting and solving for the constant which is define as the wave number, K.

$-{\omega }^2+C_o^2{\lambda }^2=0$

$K^2={\lambda }^2=\frac{{\omega }^2}{C_o^2}$

The wave number is an important quantity relating the angular velocity of the wave to its propagation speed in the medium. It can be expressed in different forms.

$K=\frac{\omega }{C_o}=\frac{2\pi f}{C_o}=\frac{2\pi }{\lambda }$

where $f$ is the frequency in hertz and $\lambda$ is the wavelength.

The second differential equation can be solved using the wave number. The spacial function is given a general form.

$X=C{e}^{jrx}$

Substituting and solving for $r$.

$-r^2+K^2=0$

$r=\pm K$

The solution of the 1-D acoustic wave equation is obtained.

$P(x,t)=(C_1{e}^{jKx}+C_2{e}^{-jKx})e^{j\omega t}$

The real and imaginary parts of the solution are also solutions to the 1-D wave equation.

$P(x,t)=C_{1}cos(\omega t+Kx)+C_{2}cos(\omega t-Kx)$

$P(x,t)=C_{1}sin(\omega t+Kx)+C_{2}sin(\omega t-Kx)$

Using phasor notation, the solution is written in more compact form.

$𝐏\mathbf{(}𝐱,𝐭\mathbf{)}=𝐏e^{j(\omega t\pm Kx)}$

The actual solution is recovered by taking the real part of the above complex form. The value of the constants above is determined by applying initial and boundary conditions. In general, any function of the following form is a solution for periodic waves.

$P(x,t)=f_1(\omega t+Kx)+f_2(\omega t-Kx)$

and similarly, for progressive waves,

$P(x,t)=f(ct+x)+g(ct-x)$

$P(x,t)=f(\xi )+g(\eta )$

where $f$ and $g$ are arbitrary functions, that represent two waves traveling in opposing directions. These are known as the d'Alembert solutions. The form of these two functions can be found by applying initial and boundary conditions.