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🌊 WAVES (Mechanical Waves)

Stage 2 – Page 4 : Standing Waves on a String (One End Fixed, One End Free)

1️⃣ Boundary Conditions

In this case:

One end is fixed → Node
Other end is free → Antinode

Golden Rule:
A free end always behaves like an antinode.

2️⃣ Allowed Standing Wave Patterns

Only those wave patterns are allowed which satisfy:

Node at fixed end
Antinode at free end

Because of this restriction, not all harmonics are possible.

3️⃣ Allowed Wavelengths

λₙ = 4L / (2n − 1)

where n = 1, 2, 3, ...

Key Observation:
Only odd harmonics are present.

4️⃣ Allowed Frequencies

fₙ = (2n − 1) × (v / 4L)

n = 1 → Fundamental frequency

5️⃣ Fundamental Mode (First Harmonic)

n = 1
λ₁ = 4L
Lowest frequency

f₁ = v / 4L

6️⃣ Comparison with String Fixed at Both Ends

Property	Both Ends Fixed	One End Free
Harmonics	All	Only odd
Fundamental λ	2L	4L
Fundamental f	v / 2L	v / 4L

7️⃣ Numerical Example

Question:
A string of length 0.5 m has one end fixed and the other free. Wave speed is 200 m/s. Find fundamental frequency.

Solution:

f₁ = v / 4L = 200 / (4 × 0.5)

f₁ = 100 Hz

8️⃣ Important JEE Points

Even harmonics are absent
Free end = maximum displacement
Frequency ratios → 1 : 3 : 5

9️⃣ Common Mistakes

Using 2L instead of 4L
Including even harmonics
Misidentifying node/antinode

🔟 Next Page Preview

Standing waves in air columns
Open pipe vs closed pipe

Stage 2 – Page 4 Completed Successfully ✅