Speed of Travelling Waves 🚀

1 • Chasing the Crest

Track a point of fixed phase on the wave—say the crest—and keep its phase constant:
\(k\,x – \omega\,t = \text{constant}\) 🙂 From this condition you get the speed \(v = \dfrac{\omega}{k}\). Because \(k = \dfrac{2\pi}{λ}\) and \(\omega = 2\pi\nu = \dfrac{2\pi}{T}\), the same speed appears in the more familiar forms \(v = λ\,\nu = \dfrac{λ}{T}\). :contentReference[oaicite:0]{index=0}

2 • What Really Sets the Speed?

Only the medium’s properties decide \(v\). A string’s tension or a gas’s bulk modulus matters—neither the amplitude nor the frequency changes the speed once the wave has started. ✨ :contentReference[oaicite:1]{index=1}

3 • Transverse Waves on a Stretched String 🎸

The tug of war is between tension \(T\) (elastic property) and linear mass density \(µ\) (inertial property). Dimensional analysis nails the relationship:
\[v = \sqrt{\dfrac{T}{µ}}\] :contentReference[oaicite:2]{index=2}

• 🔍 Example 14.3: A 0.72 m steel wire, mass \(5.0\times10^{-3}\,\text{kg}\), under \(60\;\text{N}\) tension has \(µ = 6.9\times10^{-3}\,\text{kg m}^{-1}\) and
  \(v \approx 93\;\text{m s}^{-1}\). :contentReference[oaicite:3]{index=3}

4 • Longitudinal Waves (Sound) 🔊

4.1 General Picture

The bulk modulus \(B\) plays the “spring” and density \(ρ\) plays the “mass,” giving
\[v = \sqrt{\dfrac{B}{ρ}}\] :contentReference[oaicite:4]{index=4}

4.2 In Solids

For a slender rod you swap in Young’s modulus \(Y\):
\[v = \sqrt{\dfrac{Y}{ρ}}\] :contentReference[oaicite:5]{index=5}

4.3 In Gases

If you (incorrectly) assume slow, isothermal changes you get Newton’s formula
\(v = \sqrt{\dfrac{P}{ρ}}\) —about 280 m s⁻¹ for air at STP. But sound compressions are too quick for heat flow, so the process is adiabatic. Laplace’s correction replaces \(P\) with \(γP\) (γ ≈ 7⁄5 for air):
\[v = \sqrt{\dfrac{γP}{ρ}}\] which gives 331 m s⁻¹, matching experiment. 🎉

• 🔍 Example 14.4: Using \(ρ = 1.29\;\text{kg m}^{-3}\) for air at STP and \(γ = 7/5\), the corrected formula predicts \(331\;\text{m s}^{-1}\). :contentReference[oaicite:7]{index=7}

5 • Quick Wave Check-up ✏️

Example 14.2: For \(y(x,t)=0.005\sin(80.0\,x-3.0\,t)\) (SI units) you can read off: amplitude 5 mm, wavelength \(λ = 7.85\)\,cm, period \(T = 2.09\)\,s, frequency \(ν = 0.48\)\,Hz, and after 20 s at \(x = 30\)\,cm the string is displaced ~5 mm. :contentReference[oaicite:8]{index=8}

High-Yield Nuggets for NEET 🎯

• 🔑 Formula trio: \(v = \dfrac{ω}{k} = λ\nu = \dfrac{λ}{T}\) connects speed, wavelength, period, and frequency. :contentReference[oaicite:9]{index=9}
• 🔑 Stretched-string speed: \(v = \sqrt{T/µ}\) depends only on tension and linear density—not on \(λ\) or \(ν\). :contentReference[oaicite:10]{index=10}
• 🔑 Sound speed in media: \(v = \sqrt{B/ρ}\) or \(v = \sqrt{Y/ρ}\) highlights the balance of elasticity and inertia. :contentReference[oaicite:11]{index=11}
• 🔑 Laplace correction: Use \(v = \sqrt{γP/ρ}\) (not \(P/ρ\)) for gases, because sound compressions are adiabatic. :contentReference[oaicite:12]{index=12}
• 🔑 Medium matters: High bulk (or Young’s) modulus in solids/liquids beats their larger density, making sound much faster than in gases. :contentReference[oaicite:13]{index=13}

Keep practicing, and these ideas will feel like second nature—happy learning! 😊