Superposition of Waves — Formula Sheet

$y$: Displacement of the particle at position x and time t (m) · $A$: Amplitude (m) · $n$: Frequency (Hz) · $\lambda$: Wavelength (m) · $\omega$: Angular frequency = 2πn (rad s⁻¹) · $k$: Propagation constant (wave number) = 2π/λ (rad m⁻¹)

Wave travelling along the +x direction. For a wave travelling along −x, use (nt + x/λ). The Maharashtra textbook uses n for frequency (NCERT writes ν).

$v$: Wave speed (m s⁻¹) · $T$: Period = 1/n (s)

The speed depends on the medium; the frequency is fixed by the source. When a wave enters another medium, n stays the same and λ changes. Unit: m s⁻¹.

$\Delta \varphi$: Phase difference (rad) · $\Delta x$: Path difference (m)

A path difference of one wavelength corresponds to a phase difference of 2π rad. Points λ/2 apart on a progressive wave are in opposite phase.

$A$: Amplitude of each component wave (m) · $x$: Position along the medium (m)

Formed by two identical waves travelling in opposite directions. Every particle performs SHM of frequency n, but the amplitude 2A cos(2πx/λ) changes from point to point. There is no (nt ± x/λ) term, so the disturbance does not travel.

For y = 2A cos(2πx/λ) sin(2πnt): nodes (zero amplitude) at x = λ/4, 3λ/4, 5λ/4 …, antinodes (amplitude 2A) at x = 0, λ/2, λ … Nodes and antinodes alternate.

$L$: Vibrating length of the string (m) · $T$: Tension in the string (N) · $m$: Linear density — mass per unit length (kg m⁻¹)

String fixed at both ends, vibrating in one loop. Both ends are nodes. √(T/m) is the speed of transverse waves on the string.

$p$: Number of loops (harmonic number)

String vibrating in p loops: L = pλ/2. All harmonics (n, 2n, 3n …) are present. The pth harmonic is the (p − 1)th overtone.

Law of length (T, m constant), law of tension (L, m constant) and law of linear density (L, T constant). On a sonometer with a fixed tension, nL = constant.

$v$: Speed of sound in air (m s⁻¹) · $L$: Corrected length of the air column, L = l + e (m) · $l$: Length of the pipe (m) · $e$: End correction (m)

Closed end is a node, open end is an antinode, so the simplest mode has L = λ/4. L = l + e is the corrected length of the air column; use it whenever the diameter of the pipe is given (put e = 0 only if told to ignore end correction).

$p$: Overtone number (p = 0 for the fundamental)

As in the textbook, p is the overtone number (p = 0 is the fundamental). Only odd harmonics are present: n, 3n, 5n … The first overtone (p = 1) is the 3rd harmonic, the second overtone (p = 2) is the 5th harmonic.

Both ends are antinodes, so the simplest mode has L = λ/2. End correction is applied at both open ends, so the corrected length is L = l + 2e. For the same length, an open pipe has twice the fundamental frequency of a closed pipe.

$p$: Overtone number (p = 0 for the fundamental)

p is the overtone number (p = 0 is the fundamental). All harmonics are present: n, 2n, 3n … The first overtone (p = 1) is the 2nd harmonic. Richer in harmonics than a closed pipe, so its sound quality is different.

$e$: End correction (m) · $d$: Inner diameter of the pipe (m)

d is the inner diameter of the pipe (equivalently e = 0.6r). Corrected length of the air column: L = l + e for a closed pipe, L = l + 2e for an open pipe (l = length of the pipe).

$n_1,n_2$: Fundamental frequencies of the two pipes (Hz) · $l_1,l_2$: Lengths of the two pipes (m)

Two closed pipes of the same diameter, lengths l₁ and l₂, with fundamental frequencies n₁ and n₂. Since v is the same, 4n₁(l₁ + e) = 4n₂(l₂ + e). For two pipes open at both ends the same method gives e = (n₂l₂ − n₁l₁)/[2(n₁ − n₂)].

$N$: Beats per second (Hz) · $n_1,n_2$: Frequencies of the two sources (Hz)

Number of beats per second equals the difference of the two frequencies (n₁ > n₂). Beats can be heard only when this difference is small (practically less than about 6–7 Hz for the normal human ear).

${\lambda }_1,{\lambda }_2$: Wavelengths of the two waves, λ₁ < λ₂ (m)

Two waves of wavelengths λ₁ < λ₂ in the same medium produce N beats per second. Comes from v/λ₁ − v/λ₂ = N.