Wave Equation 

The wave equation is a key mathematical model that describes how waves propagate through space and time. It’s a second-order partial differential equation that links the wave's displacement to both position and time. Assuming no energy loss, it helps analyze wave behavior in mechanical, electromagnetic, and quantum systems.

1.0Wave Function

• A wave function describes the disturbance caused by a wave. For a string, it’s the displacement, for sound waves, it’s pressure or density fluctuations, and for light, it’s the electric or magnetic field. 
• In a 1D wave traveling along the x-axis, a particle at position x is displaced by a distance y perpendicular to the x-axis. This displacement, y, depends on both position (x) and time (t), represented as y = f(x, t).This is called wave function

Different physical interpretation of wave function

• The wave function represents the displacement of a particle in the medium at a given time.
• For a fixed time $(t=t_0),y(x,t=t_0)=f(x)$ shows the wave's shape at that time.
• For a fixed position $(x=x_0),y(x=x_0,t)=f(t)$ describes the motion of the particle at  \( (x = x_0)

Wave function when equation of shape is given:

• Given a wave pulse traveling with speed v, after time t, it moves a distance vt along the +x-axis. If the pulse shape at t = 0 is y = f(x), after time t, the shape shifts by vt. Therefore, the wave function at time t is represented as y = f(x - vt).

• If the wave pulse is travelling along – x-axis then y = f(x + vt).
• Velocity of the wave,

$v=-\frac{B}{A}=\frac{Coefficientoft}{Coefficientofx}$

$y=(x-vt{)}^2,\sqrt{(x-vt)},A{e}^{-B(x-vt{)}^2}$

• Examples of Travelling Wave

$y=\left(x^2-v^2{t}^2\right),sqrtx-vt),A\sin \left(4x^2-9t^2\right)$

• Examples not Representing Travelling Wave

2.0Wave Function for a Particle's Equation of Motion

• A horizontally stretched string has a flat equilibrium shape, with displacement y measured perpendicular to the string.
• When the left end is plucked, a pulse travels rightward, with vertical displacement at x = 0 given by y(x = 0, t) = f(t).
• If there is no friction, the pulse travels undiminished at speed v, with its position at  x = vt.
• The displacement of the particle at x and time t originated at the left end at time

$t=\frac{x}{v},so(y(x,t)=f\left(t-\frac{x}{v}\right)$

$y(x,t)=y\left(x=0,t-\frac{x}{v}\right)=f\left(t-\frac{x}{v}\right)$

$\frac{f}{v}(vt-x)$

$-\frac{f}{v}(x-vt)$

• If the wave is travelling in –x direction, then wave equation is written as

$y(x,t)=f\left(t+\frac{x}{v}\right)$

• The quantity x-vt is called phase of the wave function.
• As phase of the pulse has fixed value  x-vt=Constant ,than $v=\frac{dx}{dt}$ called the phase velocity often called wave velocity.

3.0Travelling Sine wave in one Dimension

• The wave equation y = f(t-xv) or $Y=f\left(t-\frac{x}{v}\right)$ applies to both transverse and longitudinal waves with arbitrary shapes. A complete wave description requires specifying f(x). The most common case in physics and engineering is a sinusoidal wave, where f(x) is a sine or cosine function. This occurs when the source at x = 0vibrates in simple harmonic motion, continuously supplying energy to the string. Let's now consider the equation of a traveling sinusoidal wave.
• The wave equation $y=A\sin (\omega t+kx)$ implies y = 0 at x = 0 and t = 0, but this isn’t always true. To account for different initial conditions, a phase constant φ  ( \Phi) is added, giving the general form: $y=A\sin (\omega t\pm kx+\Phi )$. The phase constant $\Phi$ allows any initial condition to be satisfied.
• The equation $y(x_0,t)=A\sin (\omega t-k{x}_0)$ $y(x_0,t)=A\sin (\omega t+\Phi )$ describes simple harmonic motion (SHM) for particles at $x=x_0$ .All particles in the wave undergo SHM with the same amplitude A, angular frequency ω, and frequency f=2 .  The phase $f=\frac{\omega }{2\pi }$ depends on the particle's location. If 1 and 2 ${\varphi }_1$ and ${\varphi }_2$ are the phases of particles at positions $x_1$ and $x_2$, respectively, at time t, then,

${\Phi }_1-{\Phi }_2=k(x_2-x_1)$

=kx …………(1)

$|\Delta \Phi |=k|\Delta x|$

Shape of the wave on string at $t=t_0$

$y=A\sin (\omega t_0-kx)$

$y=A\sin (kx-\omega t_0+\pi )$

$y=A\sin (kx+\pi )\therefore \Phi =\pi -\omega t_0$

$\frac{2\pi }{k}$ is the period of the function,which is the minimum length after which wave repeats itself and is called wavelength (λ) of the wave and k is called wave number.

$\frac{|\Delta \Phi |}{2\pi }=\frac{\Delta x}{\lambda }$ ……….(3)

4.0The Linear Wave Equation

• Using the wave function $y=A\sin (\omega t-kx+\Phi )$, we can describe the vertical motion of any point on the string, with its x coordinate remaining constant. The transverse velocity $v_y$ and transverse acceleration ay any of the points are given by.

$v_y={\left[\frac{dy}{dt}\right]}_{\text{constant }x}=\frac{\partial y}{\partial t}=\omega A\cos (\omega t-kx+\Phi )$

$a_y={\left[\frac{d{v}_y}{dt}\right]}_{\text{constant }x}=\frac{\partial v_y}{\partial t}=\frac{{\partial }^{2}y}{\partial t^2}=-{\omega }^{2}A\sin (\omega t-kx+\Phi )$

$v_{y,\max }=\omega A$

$a_{y,\max }={\omega }^{2}A$

• The transverse velocity and acceleration of a point on the string don't reach their maximum values at the same time. The velocity is maximum (ωA)   (\omega A \)  when the displacement y = 0, while the acceleration is maximum (ω²A) when y = ±A.

${\left[\frac{dy}{dx}\right]}_{\text{constant }t}=\frac{\partial y}{\partial x}=-kA\cos (\omega t-kx+\Phi )$

$\frac{{\partial }^{2}y}{\partial x^2}=-k^{2}A\sin (\omega t-kx+\Phi )$

$\frac{\partial y}{\partial t}=-\omega \frac{\partial y}{\partial x}$

$v_p=-v\times \text{slope}$

Note:If the slope at a point is negative, the particle velocity is positive, and vice versa. For a wave moving along the positive x-axis, the wave velocity vw  (\v_w) is positive.

For two points A and B on the wave, moving along the positive x-axis: the slope at A is positive, so its velocity is negative, meaning it moves downward. At point B, the opposite occurs.

Differential Equation of Travelling Wave

$p,\frac{{\partial }^{2}y}{\partial x^2}=\frac{k^2}{{\omega }^2}\frac{{\partial }^{2}y}{\partial t^2}$

$\frac{{\partial }^{2}y}{\partial x^2}=\frac{1}{v^2}\frac{{\partial }^{2}y}{\partial t^2}$

5.0General Wave Expressions

1.Electromagnetic Wave

$E_y=E_0\sin (kx-\omega t)$

$B_z=B_0\sin (kx-\omega t)$

2.Displacement Waveform in Sound Wave

$s=s_0\cos (kx-\omega t)$

3.Pressure Waveform

$p=p_0\cos (kx-\omega t)$

4.Stationary Waveform 

y=2A Sin kx Sin t