Mechanical Wave on String and Sound Wave Formulas

1. Wave

When the disturbance in a medium is transmitted from one point to other but the medium itself is not transported, it is called a wave.

2. Types of waves

• Transverse
• Longitudinal.

3. Progressive wave

Waves by which energy and momentum are transported.
Equation is
y = a sin(╧ët – kx) = a sin ╧ë\(\left(t-\frac{x}{v}\right)\) = a sin 2╧Ç\(\left(\frac{t}{T}-\frac{x}{\lambda}\right)\) = a sin \(\frac{2 \pi}{\lambda}\)(vt – x)

4. Stationary waves

Waves bound in a region without transfer of energy and momentum.
When two waves of same frequency and amplitude travel in opposite directions along the same path, their superposition produces the stationary waves.

5. Characteristics of waves

v = n╬╗, n = \(\frac{1}{T}=\frac{\omega}{2 \pi}\)
k = \(\frac{2 \pi}{\lambda}=\frac{\omega}{\mathrm{v}} \Rightarrow \mathrm{v}=\frac{\omega}{\mathrm{k}}\) v → wave velocity

6. Intensity

Intensity =2╧Dzn²a²╧ü╬╜
Level of intensity or loudness ╬▓ = L = log₁₀\(\frac{1}{I_{0}}\)(dB)
I₀ is intensity of threshold of hearing = 10^-12 W/m².

7. Velocity of longitudinal waves (sound waves)

In a solid v = \(\sqrt{\frac{Y}{\rho}}\) Y is Young’s modulus and ╧ü is density.
In a fluid v = \(\sqrt{\frac{\mathrm{E}}{\rho}}\) E is Bulk modulus.
In gas v = \(\sqrt{\frac{\gamma P}{\rho}}\) E = E_adiabatic = ╬│P, ╬│ = \(\frac{C_{p}}{C_{v}}\)

8. Effect of temperature

v = \(\sqrt{\frac{\gamma \mathrm{RT}}{\mathrm{M}}}\) or v ∝ √T
\(\frac{v_{t}}{v_{0}}=\sqrt{\frac{T}{T_{0}}}=\left(\frac{273+t}{273}\right)^{1 / 2}\)
t → is temperature in °C.
v_t = v₀(1 + 0.61 t)m/s

9. Effect of pressure

No effect, as at constant T, \(\frac{P}{\rho}\) = constant.

10. Effect of Humidity

v increases with humidity as ρ decreases.

11. Interference

It is due to superposition of coherent waves. It results in redistribution of energy.
y = y₁ + y₂
= a sin ωt + b sin (ωt + Φ)
= A sin (ωt + α)
A = (a² + b² + 2ab cos ╬ª)^1/2
I = (I₁ + I₂ + 2\(\sqrt{\mathrm{I}_{1} \mathrm{I}_{2}}\)cos ╬ª)
Constructive interference, Φ = 2nπ
A = A_max = (a + b)
I = I_max = \(\left(\sqrt{I_{1}}+\sqrt{I_{2}}\right)^{2}\)
Destructive interference, (╬ª) = (2n – 1) ╧Ç
A = A_max = (a – b)
I = I_max = \(\left(\sqrt{I_{1}}-\sqrt{I_{2}}\right)^{2}\)
Phase difference Φ is equivalent to a path difference x
x = \(\frac{\lambda}{2 \pi}\)Φ.

12. Beats

Due to superposition of waves of slightly different frequencies.
Beat frequency N = Difference of frequencies = m Γëâ n

13. Stationary waves

Due to superposition of waves of same frequency traveling in opposite j direction.
Reflection from open or free end
y = a sin (╧ët – kx) + a sin (╧ët + kx)
= 2a cos x sin ωt
Reflection from closed or rigid end
y = a sin (╧ët – kx) – a sin (╧ët + kx)
= – 2a sin kx cos ╧ët

14. Nodes

Positions of minimum displacement and maximum strain, Successive nodes separated by ╬╗/2.

15. Antinodes

Positions of maximum displacement and minimum strain. Successive antinodes separated by ╬╗/2.

16. Vibrations in stretched strings

Transverse waves are formed.
Velocity = \(\sqrt{\frac{\mathrm{T}}{\mathrm{m}}}\)
T → tension in string
m → mass per unit length.
Fundamental frequency
n₁ = \(\frac{1}{2 \ell} \sqrt{\frac{T}{m}}\)
Frequency of p^th harmonic
n_p = \(\frac{P}{2 \ell} \sqrt{\frac{T}{m}}\)
n₁ : n₂ : n₃ ……….. : 1 : 2 : 3: …………..

17. Sonometer

n = \(\frac{1}{2 \ell} \sqrt{\frac{T}{m}}\)
n ∝ \(\frac{1}{\ell}\), n ∝ \(\sqrt{\mathrm{T}}\), n ∝ \(\frac{1}{\sqrt{m}}\) ∝ \(\frac{1}{\sqrt{\pi r^{2} d}}\)

18. MeldeΓÇÖs experiment

Transverse arrangement
N = n = \(\frac{\mathrm{P}}{2 \ell} \sqrt{\frac{\mathrm{T}}{\mathrm{m}}}\)
Longitudinal arrangement
N = 2n = \(\frac{P}{\ell} \sqrt{\frac{T}{m}}\)
MeldeΓÇÖs law p\(\sqrt{\mathrm{T}}\) = constant.
Vibration of string in both arrangements are transverse.

19. Vibrations of air columns (pipes)

Closed organ pipe –

Fundamental frequency n₁ = \(\frac{v}{4l}\)
First overtone = \(\frac{3v}{4l}\) = 3n₁ = Third harmonic
p^th overtone = (2p + 1)\(\frac{v}{4l}\) = (2p + 1)^th harmonic.
n₁ : n₂ : n₃ : : 1 : 3 : 5 : …………..
Only odd harmonic present.

Open organ pipe

Fundamental frequency n₁ = \(\frac{v}{2l}\)
First overtone = \(\frac{2v}{2l}\) = 2n₁ = Second harmonic
p^th overtone = (p + 1) \(\frac{v}{2l}\) = (p + 1)^th harmonic.
n₁ : n₂ : n₃ ………….. : : 1 : 2 : 3 : …………..
(All harmonics are present).

20. Resonance tube

l₁ + x = \(\frac{╬╗}{4}\), l₂ + x = \(\frac{3╬╗}{4}\)
l₁ and l₂ are first and second resonant lengths and x is end correction.
╬╗ = 2(l₂ – l₁),v = 2n(l₂ – l₁)
End correction
x = \(\frac{\left(\ell_{2}-3 \ell_{1}\right)}{4}\) = 0.6r = 0.3d

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