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🌊 WAVES (Mechanical Waves)

Stage 2 – Page 2 : Wave on a Stretched String (Numericals & Concepts)

1️⃣ Physical Meaning of Wave on String

A wave on a stretched string propagates due to tension and mass distribution of the string.

Key Idea:
Higher tension → faster wave Heavier string → slower wave

2️⃣ Fundamental Formula (Most Important)

v = √(T / μ)

Symbol	Meaning
v	Wave speed (m/s)
T	Tension (N)
μ	Linear mass density (kg/m)

3️⃣ Linear Mass Density (μ)

Linear mass density is defined as:

μ = m / L

where m = mass of string and L = length of string.

Exam Trap:
Do NOT use total mass directly — always convert to μ.

4️⃣ Numerical Example 1

Question:
A string of length 5 m and mass 0.5 kg is stretched with tension 200 N. Find wave speed.

Solution:

μ = m / L = 0.5 / 5 = 0.1 kg/m

v = √(T / μ) = √(200 / 0.1)

v = √2000 ≈ 44.7 m/s

5️⃣ Effect of Changing Parameters

Change	Effect on Speed
T → 4T	v → 2v
μ → 4μ	v → v/2

6️⃣ Frequency Relation on String

Using v = fλ:

f = (1/λ) √(T / μ)

Useful when wavelength is fixed by boundary conditions.

7️⃣ Typical JEE Numerical – Concept Based

Question:
If tension in a string is doubled, how does frequency change?