Superposition of Waves — MHT-CET Physics Formulas and Traps

Progressive Waves: the Wave Equation, Phase and Particle Velocity

Reading the Wave Equation

Progressive wave

y = A\sin(\omega t - kx), \qquad v = \frac{\omega}{k} = f\lambda

Phase Difference and Path Difference

Phase and path

\Delta\phi = \frac{2\pi}{\lambda}\,\Delta x

Particle Velocity Against Wave Velocity

Speed ratio

\frac{v_{p,\max}}{v_{\text{wave}}} = Ak = \frac{2\pi A}{\lambda}

Common traps

Leaving π in the wave number

In

12\pi t - 0.02\pi x

,

k = 0.02\pi

, so

v = 600

m/s. Reading

k = 0.02

gives

600\pi

— watch whether the

\pi

is attached to x.

Mixing degrees and radians

\Delta x = \frac{\lambda}{2\pi}\Delta\phi

needs

\Delta\phi

in radians; in degrees use

\frac{\Delta\phi}{360^\circ}\lambda

. Plugging 60 into the radian form is the classic slip.

Treating a sin² wave like a sine wave

A\sin^2\theta = \frac{A}{2}(1 - \cos 2\theta)

: the oscillation has amplitude

\frac{A}{2}

at twice the frequency, so the particles peak at

A\omega

, not

2A\omega

— and the wavelength is

\frac{\pi}{k}

.

Superposition of Two Waves

Learn this subtopic in the notes

The Resultant Amplitude

Resultant amplitude

R^2 = a_1^2 + a_2^2 + 2a_1a_2\cos\phi

Resultant Intensity and Mixed Sine–Cosine Waves

Resultant intensity

I = I_1 + I_2 + 2\sqrt{I_1I_2}\cos\phi

Common traps

Adding amplitudes at 90°

Two waves a quarter-cycle apart do not give 2A; they give

\sqrt{2}A

. Only waves in step add their amplitudes directly.

Reading φ off a sine–cosine pair

a_1\sin(\omega t - kx)

and

a_2\cos(\omega t - kx + \phi)

differ by

\phi + \frac{\pi}{2}

, not

\phi

. Convert the cosine first.

Stationary Waves and Vibrating Strings

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Nodes, Antinodes and the Standing-Wave Equation

Stationary wave

y = 2A\sin kx\cos\omega t, \qquad \text{node spacing} = \frac{\lambda}{2}

Frequencies of a Stretched String

String harmonics

n_p = \frac{p}{2L}\sqrt{\frac{T}{\mu}}

Common traps

Taking the node spacing as λ

Nodes are HALF a wavelength apart. From

\sin\frac{\pi x}{4}

,

\lambda = 8

cm but the nodes are 4 cm apart.

Frequency proportional to tension

n \propto \sqrt{T}

. A 44% rise in tension raises the frequency by 20% (

\sqrt{1.44} = 1.2

), not 44%.

Organ Pipes, the Resonance Tube and the Doppler Effect

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Open and Closed Pipes

Pipe frequencies

n_{\text{open}} = \frac{pv}{2L}, \qquad n_{\text{closed}} = \frac{(2p - 1)v}{4L}

The Doppler Effect

Doppler effect

n = n_0\,\frac{v + v_L}{v - v_S} \quad (\text{approaching})

Common traps

Numbering a closed pipe's overtones like an open one's

A closed pipe has no even harmonics, so its first overtone is the 3RD harmonic and its second the 5th. Counting 2nd, 3rd as for an open pipe gives the wrong frequency every time.

One end correction for an open pipe

An open pipe has TWO open ends, so its effective length is

l + 2e

; a closed pipe has one,

l + e

. Using

l + e

for an open pipe is the planted wrong wavelength.

Putting the source's speed in the numerator

The LISTENER's speed goes on top, the SOURCE's underneath. Approaching means plus on top and minus below — both push the pitch up.

Beats, and Tuning a Sonometer

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Beat Frequency, Timing and Loudness

Beats

f_{\text{beat}} = |f_1 - f_2|, \qquad T_{\text{beat}} = \frac{1}{|f_1 - f_2|}

A Tuning Fork Against a Sonometer

Sonometer

f \propto \frac{1}{l}\;\Rightarrow\; f_1l_1 = f_2l_2

Common traps

Reading ω as the frequency

In

\sin 316t

, 316 is ω, not f. The beat frequency is

\frac{316 - 310}{2\pi} = \frac{3}{\pi}

, not 6.

Pairing the higher frequency with the longer length

A SHORTER wire vibrates faster. Solve

f_1l_1 = f_2l_2

with the larger f on the smaller l, or the two frequencies come out swapped.

Superposition of Waves formulas

Progressive Waves: the Wave Equation, Phase and Particle Velocity

Reading the Wave Equation

Phase Difference and Path Difference

Particle Velocity Against Wave Velocity

Common traps

Superposition of Two Waves

The Resultant Amplitude

Resultant Intensity and Mixed Sine–Cosine Waves

Common traps

Stationary Waves and Vibrating Strings

Nodes, Antinodes and the Standing-Wave Equation

Frequencies of a Stretched String

Common traps

Organ Pipes, the Resonance Tube and the Doppler Effect

Open and Closed Pipes

The Doppler Effect

Common traps

Beats, and Tuning a Sonometer

Beat Frequency, Timing and Loudness

A Tuning Fork Against a Sonometer

Common traps

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