Wave Equation Solutions

• A common approach to solving differential equations is to assume sinusoidal solutions in the form of complex exponentials.

• A time-harmonic representation of displacement can be expressed as $y(x, t) = \tilde{y}(x) e^{j \omega t}$, where $\omega$ is radian frequency and $\tilde{y}(x)$ represents the spatial distribution of the complex amplitude displacement.

• Substituting this expression into the wave equation yields the 1D form of the Helmholtz equation
    $$\partial^{2} \tilde{y} / \partial x^{2} + (\omega / c)^2 \tilde{y}(x) = 0,$$ (2)
  which is a linear, second-order, ordinary differential equation (a function of $x$ alone).

• A standard trial solution for Eq. (2) is of the form $\tilde{y}(x) = \tilde{A} e^{\lambda x}$, which after substitution yields:
  $$(\lambda^2 + k^2)\tilde{y}(x) = 0,$$
  where $\lambda$ is the wavelength and $k = \omega / c = 2 \pi / \lambda$ is the wavenumber, which represents spatial frequency in radians per meter.

• There are two non-trivial solutions of $\lambda = \pm j k$, leading to a complete solution of the form:
  $$y(x, t) = C^{+} e^{j(\omega t - k x)} + C^{-} e^{j(\omega t + k x)},$$
  where $C^{+}$ and $C^{-}$ are complex amplitudes.

• In terms of sinusoidal functions, this can expressed as:
  

  $$y(x,t) = A \sin \frac{\omega}{c} (ct - x) + B \cos \frac{\omega}{c} (ct - x) + C \sin \frac{\omega}{c} (ct + x) + D \cos \frac{\omega}{c} (ct + x)$$

  $$= A \sin (\omega t - kx) + B \cos (\omega t - kx) + C \sin (\omega t + kx) + D \cos (\omega t + kx).$$ (3)

• This solution can be represented in a more general form, attributed to d'Alembert in 1747, of $y(x,t) = y^{+}(ct - x) + y^{-}(ct + x)$.

• $y^{+}(ct - x)$ represents a wave traveling in the positive $x$ direction with a velocity $c$. Similarly, $y^{-}(ct + x)$ represents a wave traveling in the negative $x$ direction with the same velocity. Each component is generally referred to as a traveling wave.

• The functions $y^{+}$ and $y^{-}$ are arbitrary and of fixed shape (given our assumed lossless medium) ... see waves on string simulation.

• This implies that waves can propagate in two opposite directions in a one-dimensional medium.

  Figure 2: Traveling waves on a string.

  

• When two or more waves pass through the same region of space at the same time, the actual displacement is the vector (or algebraic) sum of the individual displacements (referred to as linear superposition).