Waves on a String – Wave Speed, Modes, Standing Waves & Formulas | Entrance Exam Physics

WAVES ON A STRING — SHORT NOTES

1. Wave Speed on a Stretched String

Wave speed depends on tension (T) and linear mass density (μ):

$$v=\sqrt{\frac{T}{\mathrm{\mu }}}$$

where

$$\mu =\frac{m}{L}$$

✔ Higher tension → faster waves
✔ Heavier string → slower waves

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2. Types of Waves on a String

Transverse Waves

Particles vibrate ⟂ to the direction of wave motion.

General equation:

$$y(x,t)=A\sin (kx-\omega t)$$

Where

$$k=\frac{2\pi }{\lambda },\omega =2\pi f$$

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3. Relation Between Speed, Wavelength, Frequency

$$v=f\lambda$$

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4. Standing Waves on a String

Formed when incident and reflected waves interfere (at fixed ends).

Boundary Conditions

• Fixed end: displacement = 0 (node)

• Both ends fixed: N–A–N pattern

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5. Allowed Wavelengths on a String

$${\lambda }_n=\frac{2L}{n};n=1,2,3\dots$$

(Only certain wavelengths satisfy node–antinode conditions.)

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6. Natural Frequencies (Harmonics)

$$f_n=\frac{nv}{2L}=\frac{n}{2L}\sqrt{\frac{T}{\mu }}$$

n = 1 (Fundamental)

$$f_1=\frac{1}{2L}\sqrt{\frac{T}{\mu }}$$

n = 2 (1st Overtone)

$$f_2=\frac{2}{2L}\sqrt{\frac{T}{\mu }}$$

n = 3 (2nd Overtone)

$$f_3=\frac{3}{2L}\sqrt{\frac{T}{\mu }}$$

✔ Frequencies are in simple integer ratio 1 : 2 : 3 …

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7. Node–Antinode Pattern

• Distance between two nodes = λ/2

• Distance between node and adjacent antinode = λ/4

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8. Energy Transport

Average power transmitted:

$$P=\frac{1}{2}\mu v{\omega }^2{A}^2$$

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9. Effect of Changing Parameters

• Increase tension ⇒ frequency ↑

• Increase length ⇒ frequency ↓

• Increase mass density ⇒ frequency ↓

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10. Common Entrance Exam Problems

• Using 

  $v=\sqrt{T/\mu}$ to find tension

• Calculating harmonic frequencies

• Determining node/antinode positions

• Adjusting tension for desired frequency

• Comparing frequencies of two different strings

• Fundamental vs overtone questions