Superposition of Waves

Flashcards on wave properties, progressive and stationary waves, superposition principles, reflection conditions, and vibrating air columns.

What is a progressive wave (or travelling wave)?

A progressive wave is a wave in which the disturbance travels continuously in one direction, without any damping and obstruction, from one particle to another.

What are the primary differences in particle vibration between transverse waves and longitudinal waves?

In a transverse wave, particle vibrations are perpendicular to the direction of wave propagation, producing crests and troughs. In a longitudinal wave, particle vibrations are along the direction of wave propagation, producing compressions and rarefactions.

What is the displacement equation for a simple harmonic progressive wave travelling in the positive $x$-direction?

$y(x,t) = A\thinspace\text{sin}(kx - \text{ω}t)$, where $A$ is the amplitude, $k = \frac{2\text{π}}{\text{λ}}$ is the wave number, and $\text{ω}$ is the angular frequency.

How does a transverse wave reflect from a rigid boundary (denser medium)?

When a transverse wave reflects from a rigid boundary, a crest is reflected as a trough and a trough is reflected as a crest, resulting in a phase change of $\text{π}\thinspace\text{radian}$ ($180^°$).

How does a transverse wave reflect from a flexible boundary (rarer medium)?

When a transverse wave reflects from a flexible boundary, a crest is reflected back as a crest and a trough as a trough, with no phase change.

How does a longitudinal wave reflect when travelling from a rarer medium to a denser medium?

When a longitudinal wave travels from a rarer medium to a denser medium, a compression is reflected as a compression and a rarefaction is reflected as a rarefaction, with no change of pressure phase.

How does a longitudinal wave reflect when travelling from a denser medium to a rarer medium?

When a longitudinal wave travels from a denser medium to a rarer medium, a compression is reflected as a rarefaction and a rarefaction as a compression, with a pressure phase change of $\text{π}\thinspace\text{radians}$.

What is the Principle of Superposition of Waves?

When two or more waves travelling through a medium pass through a common point, each wave produces its own displacement at that point independent of the presence of the other wave. The resultant displacement at that point is equal to the vector sum of the displacements due to the individual waves.

What is antisound and how is it used in active noise cancellation headphones?

Antisound consists of sound waveforms created by a special electronic circuit that are exactly opposite in phase to detected ambient noise. When added in headphones, the displacements cancel the noise through destructive interference.

What is the formula for the resultant amplitude $A$ of two superposed waves with amplitudes $A_1$ and $A_2$ differing in phase by $\text{φ}$?

$$A = \text{√}(A_1^2 + 2A_1 A_2\thinspace\text{cos}(\text{φ}) + A_2^2)$$

Under what conditions do superposed waves produce maximum and minimum resultant amplitudes?

Maximum amplitude $A = A_1 + A_2$ occurs when the waves interfere in phase ($\text{φ} = 0$). Minimum amplitude $A = |A_1 - A_2|$ occurs when the waves interfere out of phase ($\text{φ} = \text{π}$).

How is a stationary (standing) wave formed?

A stationary wave is formed when two identical progressive waves (having the same amplitude and frequency) travelling along the same path in opposite directions interfere with each other.

What is the displacement equation of a stationary wave on a stretched string?

y = 2a\thinspace\text{cos}\text{∶}\right(\frac{2\text{π}x}{\text{λ}}\right)\thinspace\text{sin}(2\text{π}nt) = A\thinspace\text{sin}(2\text{π}nt), where A = 2a\thinspace\text{cos}\text{∶}\right(\frac{2\text{π}x}{\text{λ}}\right) represents the position-dependent amplitude.

What are nodes in a stationary wave, and what condition determines their positions?

Nodes are points of minimum (zero) displacement. Their positions are given by $x = (2p - 1)\frac{\text{λ}}{4}$, where $p = 1, 2, 3, …$.

What are antinodes in a stationary wave, and what condition determines their positions?

Antinodes are points of maximum displacement ($\text{±}2a$). Their positions are given by $x = \frac{p\text{λ}}{2}$, where $p = 0, 1, 2, 3, …$.

What are the distances between successive nodes, successive antinodes, and an adjacent node and antinode in a stationary wave?

The distance between two successive nodes is $\frac{\text{λ}}{2}$, between two successive antinodes is $\frac{\text{λ}}{2}$, and between a node and an adjacent antinode is $\frac{\text{λ}}{4}$.

What are two key differences between progressive waves and stationary waves?

In progressive waves, disturbance/energy is transmitted with a definite velocity and all particles have the same amplitude. In stationary waves, energy is localized (no transfer, velocity is zero) and amplitudes vary from zero at nodes to maximum at antinodes.

What is the difference between harmonics and overtones?

Harmonics are frequencies that are integral multiples of the fundamental frequency ($f, 2f, 3f, …$), with the fundamental frequency being the first harmonic. Overtones are higher frequencies arranged sequentially above the fundamental frequency, which may or may not be harmonics.

What is end correction $e$ in an air column pipe, and what is its formula according to Reynold?

End correction $e$ is the distance between the open end of a pipe and the position of the antinode. According to Reynold, $e = 0.3d$, where $d$ is the inner diameter of the pipe.

What are the corrected lengths $L$ of air columns in a pipe closed at one end versus a pipe open at both ends?

For a pipe closed at one end, $L = l + e$. For a pipe open at both ends, $L = l + 2e$, where $l$ is the length of the pipe and $e$ is the end correction.

Which harmonics are present in an air column vibrating in a pipe closed at one end?

Only odd harmonics ($n, 3n, 5n, …$) are present in a pipe closed at one end; even harmonics are absent. The frequency of the $p^{\text{th}}$ overtone is given by $n_p = (2p + 1)n$.