Wave Equation — Revision Notes

⚡ 30-Second Revision

General Wave Equation: — $y(x,t) = A \sin(kx \pm \omega t + \phi)$
Amplitude: — $A$ (max displacement)
Angular Wave Number: — $k = 2\pi/\lambda$
Wavelength: — $\lambda$
Angular Frequency: — $\omega = 2\pi f = 2\pi/T$
Frequency: — $f$
Time Period: — $T$
Wave Speed: — $v = f\lambda = \omega/k$
Direction of Propagation: — $(kx - \omega t)$ for +x, $(kx + \omega t)$ for -x
Phase Difference (spatial): — $\Delta\Phi = k \Delta x = (2\pi/\lambda) \Delta x$
Phase Difference (temporal): — $\Delta\Phi = \omega \Delta t = (2\pi/T) \Delta t$
Differential Wave Equation (1D): — $\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}$
Wave Speed in String: — $v = \sqrt{T/\mu}$
Wave Speed in Fluid (Sound): — $v = \sqrt{B/\rho}$
Medium Change: — Frequency ($f$) remains constant.

2-Minute Revision

The wave equation is a mathematical model for propagating disturbances. The most common form is $y(x,t) = A \sin(kx \pm \omega t + \phi)$. Here, $A$ is amplitude, $k$ is angular wave number ($2\pi/\lambda$), $\omega$ is angular frequency ($2\pi f$), and $\phi$ is the initial phase.

The sign in $kx \pm \omega t$ determines the direction: minus for positive x-direction, plus for negative x-direction. Key relationships are $v = f\lambda = \omega/k$. Remember that wave speed ($v$) depends only on the medium, not on the source or amplitude.

When a wave passes from one medium to another, its frequency ($f$) remains constant, while its speed ($v$) and wavelength ($\lambda$) change. Phase difference between two points separated by $\Delta x$ is $\Delta\Phi = (2\pi/\lambda) \Delta x$, and for two times separated by $\Delta t$ at the same point, it's $\Delta\Phi = (2\pi/T) \Delta t$.

The differential wave equation $\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}$ is a fundamental description, and any function of the form $f(x \pm vt)$ is a solution.

5-Minute Revision

Let's consolidate the wave equation concepts for NEET. A wave is a disturbance transferring energy without matter. Its mathematical description is typically $y(x,t) = A \sin(kx \pm \omega t + \phi)$.

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Parameters:

* **Amplitude ($A$):** Maximum displacement. Energy $\propto A^2$. * **Angular Wave Number ($k$):** $k = 2\pi/\lambda$. Spatial periodicity. Units: rad/m. * **Wavelength ($\lambda$):** Distance for one cycle.

Units: m. * **Angular Frequency ($\omega$):** $\omega = 2\pi f = 2\pi/T$. Temporal periodicity. Units: rad/s. * **Frequency ($f$):** Number of cycles per second. Units: Hz. * **Time Period ($T$):** Time for one cycle.

Units: s. * **Wave Speed ($v$):** $v = f\lambda = \omega/k$. Depends on medium only. * **Phase Constant ($\phi$):** Initial phase at $x=0, t=0$.

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Direction of Propagation:

* $y(x,t) = A \sin(kx - \omega t + \phi)$: Wave moves in positive x-direction. * $y(x,t) = A \sin(kx + \omega t + \phi)$: Wave moves in negative x-direction.

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Phase Difference:

* Between two points separated by $\Delta x$: $\Delta\Phi = k \Delta x = (2\pi/\lambda) \Delta x$. * At a single point over time interval $\Delta t$: $\Delta\Phi = \omega \Delta t = (2\pi/T) \Delta t$. * Example: If $\Delta x = \lambda/4$, then $\Delta\Phi = (2\pi/\lambda)(\lambda/4) = \pi/2,\text{rad}$.

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Change of Medium: — When a wave crosses a boundary into a new medium, its **frequency ($f$) remains constant**. Its speed ($v$) and wavelength ($\lambda$) change according to the new medium's properties. So, $v_1 = f\lambda_1$ and $v_2 = f\lambda_2$.

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Differential Wave Equation: — $\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}$. This is the fundamental equation. Any function of the form $f(x \pm vt)$ is a solution, representing a wave propagating without distortion.

Worked Example: A wave is given by $y(x,t) = 0.04 \sin(10x - 50t)$. Find its amplitude, wavelength, frequency, and speed.

Amplitude (A): — Comparing with $A \sin(kx - \omega t)$, $A = 0.04,\text{m}$.
Angular Wave Number (k): — $k = 10,\text{rad/m}$. Wavelength $\lambda = 2\pi/k = 2\pi/10 = \pi/5,\text{m}$.
Angular Frequency ($\omega$): — $\omega = 50,\text{rad/s}$. Frequency $f = \omega/2\pi = 50/2\pi = 25/\pi,\text{Hz}$.
Wave Speed (v): — $v = \omega/k = 50/10 = 5,\text{m/s}$. Alternatively, $v = f\lambda = (25/\pi) \times (\pi/5) = 5,\text{m/s}$.

Master these relationships and parameter identifications for quick problem-solving.

Prelims Revision Notes

The wave equation is a core concept for NEET UG Physics, describing how disturbances propagate. The most commonly tested form is the sinusoidal traveling wave: $y(x,t) = A \sin(kx \pm \omega t + \phi)$.

Key Parameters and Formulas:

Amplitude ($A$): — Maximum displacement from equilibrium. Units: meters (m).
Angular Wave Number ($k$): — $k = \frac{2\pi}{\lambda}$. Represents spatial variation. Units: rad/m.
Wavelength ($\lambda$): — Spatial period, distance between two consecutive crests/troughs. Units: meters (m).
Angular Frequency ($\omega$): — $\omega = 2\pi f = \frac{2\pi}{T}$. Represents temporal variation. Units: rad/s.
Frequency ($f$): — Number of oscillations per second. Units: Hertz (Hz).
Time Period ($T$): — Time for one complete oscillation. Units: seconds (s).
Wave Speed ($v$): — $v = f\lambda = \frac{\omega}{k}$. This speed depends *only* on the properties of the medium, not on the source or amplitude.

Direction of Propagation:

If the phase is $(kx - \omega t)$, the wave propagates in the positive x-direction.
If the phase is $(kx + \omega t)$, the wave propagates in the negative x-direction.

Phase Difference:

Spatial Phase Difference: — For two points separated by $\Delta x$ at the same time: $\Delta\Phi = k \Delta x = \frac{2\pi}{\lambda} \Delta x$.
Temporal Phase Difference: — For the same point at two different times separated by $\Delta t$: $\Delta\Phi = \omega \Delta t = \frac{2\pi}{T} \Delta t$.

Behavior on Changing Medium:

When a wave travels from one medium to another, its **frequency ($f$) remains constant**. This is because frequency is determined by the source.
The wave speed ($v$) and wavelength ($\lambda$) *change* according to the properties of the new medium. The relationship $v = f\lambda$ still holds.

Differential Wave Equation:

The fundamental 1D wave equation is $\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}$.
Any function of the form $f(x \pm vt)$ is a solution to this equation, representing a wave that propagates without changing its shape.

Common Traps:

Confusing $k$ with $\lambda$ or $\omega$ with $f$ directly without the $2\pi$ factor.
Incorrectly determining the direction of propagation.
Assuming frequency changes when a wave enters a new medium.
Mixing up units (e.g., cm vs. m). Always ensure consistency.

Vyyuha Quick Recall

To remember the relationships between wave parameters: 'V-F-L' for $V = F\lambda$ (Velocity = Frequency x Lambda). For angular terms, think 'K-W-V' for $V = \omega/K$ (Velocity = Omega / K). And always remember '2-Pi-K' for $\lambda = 2\pi/K$ and '2-Pi-F' for $T = 2\pi/\omega$ (or $\omega = 2\pi f$).

For direction: 'Minus Means Move Forward' (kx - \omega t means +x direction).