Class 11 > Unit # 12:Acoustics > Stationary Waves (Nodes & Antinodes)

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Stationary Waves (Positions of Nodes & Antinodes) - Unit #12 Acoustics

Video Lecture: Stationary Waves & Wave Patterns

Watch the complete lecture detailing stationary waves, real-world applications like microwave ovens, and the positions of nodes and antinodes:

Introduction to Stationary (Standing) Waves

Have you ever wondered why food inside a microwave oven rotates on a turntable? Microwaves create stationary (standing) wave patterns inside the cavity where certain positions experience intense heating while other positions experience minimal heating. To ensure even cooking, the food plate must rotate.

1. What is a Stationary Wave?

A stationary wave (or standing wave) is a wave that oscillates in place within a medium and does not appear to travel through space. It is typically formed by the interference of two identical waves traveling in opposite directions—often created when an original incident wave reflects off a rigid boundary.

2. Mathematical Derivation of the Resultant Wave

Consider an incident wave traveling in the positive direction and a reflected wave traveling in the negative direction, both sharing identical amplitude $a_0$, frequency, and wavelength:

$y_1 = a_0 \sin(kx + \omega t) \quad \text{(Incident Wave)}$

$y_2 = a_0 \sin(kx - \omega t) \quad \text{(Reflected Wave)}$

Applying the superposition principle, the resultant displacement $y$ is the sum of both individual wave displacements:

$y = y_1 + y_2 = a_0 [\sin(kx + \omega t) + \sin(kx - \omega t)]$

Using trigonometric sum-to-product identities ($\sin \alpha + \sin \beta = 2 \sin\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)$), the expression simplifies to:

$y = 2a_0 \sin(kx) \cos(\omega t)$

This equation represents a stationary wave. The term $2a_0 \sin(kx)$ acts as the position-dependent amplitude of the wave.

3. Positions of Nodes (Points of Zero Displacement)

Nodes are points along a stationary wave where the amplitude of oscillation is permanently zero. This occurs when the position-dependent amplitude term is zero:

$\sin(kx) = 0 \implies kx = n\pi \quad (\text{where } n = 0, 1, 2, 3, \dots)$

Substituting the wave number $k = \frac{2\pi}{\lambda}$, we solve for the position $x$ of the nodes:

$x = \frac{n\lambda}{2} \quad \text{where } n = 0, 1, 2, 3, \dots$

Thus, nodes occur at intervals of half-wavelengths: $0, \frac{\lambda}{2}, \lambda, \frac{3\lambda}{2}, 2\lambda, \dots$

4. Positions of Antinodes (Points of Maximum Displacement)

Antinodes are points along a stationary wave where the oscillation amplitude reaches its maximum value. This occurs when:

$\sin(kx) = \pm 1 \implies kx = \left(n + \frac{1}{2}\right)\pi \quad (\text{where } n = 0, 1, 2, 3, \dots)$

Solving for position $x$ yields:

$x = \left(n + \frac{1}{2}\right)\frac{\lambda}{2} = \frac{\lambda}{4}, \frac{3\lambda}{4}, \frac{5\lambda}{4}, \dots$

Thus, antinodes occur at odd multiples of quarter-wavelengths.

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