General Equation of Wave Motion:

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

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

where $y(x, t)$ should be finite everywhere.

$\Rightarrow \quad f\left(t + \frac{x}{v}\right)$ represents a wave traveling in the $-x$-axis direction.

$\Rightarrow \quad f\left(t - \frac{x}{v}\right)$ represents a wave traveling in the $+x$-axis direction.

$$ y = A \sin (\omega t \pm k x + \phi) $$

Terms Related to Wave Motion (For 1-D Progressive Sine Wave):

• Wave number (or propagation constant) ($k$):

$$ k = \frac{2 \pi}{\lambda} = \frac{\omega}{v} \quad (\text{rad m}^{-1}) $$

• Phase of wave: The argument of the harmonic function ($\omega t \pm k x + \phi$) is called the phase of the wave.

• Phase difference ($\Delta \phi$): difference in phases of two particles at any time $t$.

$$ \Delta \phi = \frac{2 \pi}{\lambda} \Delta x \quad \text{Also,} \quad \Delta \phi = \frac{2 \pi}{T} \Delta t $$

Speed of Transverse Wave Along a String/Wire:

$$ v = \sqrt{\frac{T}{\mu}} \quad \text{where} \quad T = \text{Tension}, \quad \mu = \text{mass per unit length} $$

Power Transmitted Along the String by a Sine Wave:

• Average Power: $\langle P \rangle = 2 \pi^{2} f^{2} A^{2} \mu V$

• Intensity: $I = \frac{\langle P \rangle}{s} = 2 \pi^{2} f^{2} A^{2} \rho V$

Reflection and Refraction of Waves:

$$y_{i} = A_{i} \sin \left(\omega t - k_{1} x\right)$$

• If incident from rarer to denser medium ($v_{2} < v_{1}$):

$$y_{t} = A_{t} \sin \left(\omega t - k_{2} x\right)$$

$$y_{r} = -A_{r} \sin \left(\omega t + k_{1} x\right)$$

• If incident from denser to rarer medium ($v_{2} > v_{1}$):

$$y_{t} = A_{t} \sin \left(\omega t - k_{2} x\right)$$

$$y_{r} = A_{r} \sin \left(\omega t + k_{1} x\right)$$

• Amplitude of reflected and transmitted waves:

$$A_{r} = \frac{\left|k_{1} - k_{2}\right|}{k_{1} + k_{2}} A_{i}$$

$$A_{t} = \frac{2 k_{1}}{k_{1} + k_{2}} A_{i}$$

Standing/Stationary Waves:

$$y_{1} = A \sin \left(\omega t - k x + \theta_{1}\right)$$

$$y_{2} = A \sin \left(\omega t + k x + \theta_{2}\right)$$

$$y_{1} + y_{2} = \left[2 A \cos \left(k x + \frac{\theta_{2} - \theta_{1}}{2}\right)\right] \sin \left(\omega t + \frac{\theta_{1} + \theta_{2}}{2}\right)$$

The quantity $2 A \cos \left(k x + \frac{\theta_{2} - \theta_{1}}{2}\right)$ represents the resultant amplitude at $x$. At some positions, the resultant amplitude is zero; these are called nodes. At some positions, the resultant amplitude is $2A$; these are called antinodes.

• Distance between successive nodes or antinodes $= \frac{\lambda}{2}$.

• Distance between successive nodes and antinodes $= \frac{\lambda}{4}$.

• All the particles in the same segment (portion between two successive nodes) vibrate in the same phase.

• The particles in two consecutive segments vibrate in opposite phase.

• Since nodes are permanently at rest, energy cannot be transmitted across them.

Vibrations of Strings (Standing Wave):

$\textbf{(a) Fixed at both ends:}$

• Fixed ends will be nodes. So waves for which $L = \frac{\lambda}{2}$, $L = \frac{2\lambda}{2}$, $L = \frac{3\lambda}{2}$, etc.are possible, giving $$L = \frac{n\lambda}{2} \quad \text{or} \quad \lambda = \frac{2L}{n} \quad \text{where} \quad n = 1, 2, 3, \ldots$$

$$\text{As} \quad v = \sqrt{\frac{T}{\mu}} \quad f_{n} = \frac{n}{2L} \sqrt{\frac{T}{\mu}}, \quad n = \text{number of loops}$$

$\textbf{(b) String free at one end:}$

• For the fundamental mode, $$L = \frac{\lambda}{4} \text{or} \lambda = 4L.$$

• First overtone: $$L = \frac{3\lambda}{4},$$

hence $$\lambda = \frac{4L}{3},$$

so $$f_{1} = \frac{3}{4L} \sqrt{\frac{T}{\mu}}.$$

• Second overtone: $$f_{2} = \frac{5}{4L} \sqrt{\frac{T}{\mu}},$$

so $$f_{n} = \frac{(n + \frac{1}{2})}{2L} \sqrt{\frac{T}{\mu}} = \frac{(2n + 1)}{4L} \sqrt{\frac{T}{\mu}}.$$

Doppler Effect for Sound:

• Source moving towards the stationary observer: $$f’ = \frac{f}{1 - \frac{v_s}{v}}$$
• Source moving away from the stationary observer: $$f’ = \frac{f}{1 + \frac{v_s}{v}}$$
• Observer moving towards the stationary source: $$f’ = f \left(1 + \frac{v_o}{v}\right)$$
• Observer moving away from the stationary source: $$f’ = f \left(1 - \frac{v_o}{v}\right)$$