Unit 8: Wave in Pipes and Strings

Introduction

Standing waves, or stationary waves, arise when two waves of equal frequency and amplitude travel in opposite directions and interfere. In musical acoustics, they are essential for understanding the sound produced by wind instruments (pipes) and stringed instruments. This unit examines the physics behind these phenomena, focusing on the conditions for node and antinode formation, harmonic frequencies, end corrections, and the laws that govern string vibration.

1. Stationary Waves in Pipes

In a pipe, the boundary conditions at the ends dictate whether a node or an antinode forms. A closed end must be a node (zero displacement), while an open end must be an antinode (maximum displacement).

Closed Pipe (One end closed, one end open)

Node at the closed end, antinode at the open end.
The simplest standing wave (fundamental) fits a quarter wavelength inside the pipe:
L = λ/4 → f = v / (4L)
Only odd harmonics are present because higher modes must also satisfy the node‑antinode condition.
Overtones: 3rd harmonic (3f), 5th harmonic (5f), 7th harmonic (7f), …

Standing wave in a closed pipe showing node at closed end, antinode at open end — Figure 1: Standing wave pattern in a closed pipe (fundamental and first overtone).

Open Pipe (Both ends open)

Antinode at both ends.
The fundamental fits half a wavelength:
L = λ/2 → f = v / (2L)
All integer harmonics (both even and odd) are allowed.
Overtones: 2nd harmonic (2f), 3rd harmonic (3f), 4th harmonic (4f), …

Standing wave in an open pipe showing antinodes at both ends — Figure 2: Standing wave pattern in an open pipe (fundamental and first two overtones).

2. Harmonics and Overtones

The relationship between harmonics and overtones is simple: an overtone is any resonant frequency above the fundamental. The first overtone corresponds to the second harmonic, the second overtone to the third harmonic, and so on. Mathematically:

Overtone number = Harmonic number – 1

Pipe Type	Fundamental (f)	Harmonics Present	Overtones (first four)
Closed	`f = v/(4L)`	f, 3f, 5f, 7f, … (odd only)	3f, 5f, 7f, 9f
Open	`f = v/(2L)`	f, 2f, 3f, 4f, … (all)	2f, 3f, 4f, 5f

3. End Correction

Because the air at an open end does not exactly terminate at the physical opening, the effective length of the pipe is slightly longer than its measured length. This extra length is called the end correction (e). For a cylindrical pipe of radius r:

e ≈ 0.61 r

The effective length (L_eff) used in the resonance formulas becomes:

Closed pipe: L_eff = L + e
Open pipe: L_eff = L + 2e (correction at both ends)

Example: A pipe of radius 2 cm has e = 0.61 × 2 cm ≈ 1.22 cm. If its measured length is 30 cm, the effective length for an open pipe is L_eff = 30 cm + 2×1.22 cm ≈ 32.44 cm.

4. Velocity of Transverse Waves on a String

The speed of a transverse wave traveling along a taut string depends on the tension (T) and the linear mass density (μ, mass per unit length):

v = √(T/μ)

Where:

T = tension in the string (newtons)
μ = mass per unit length (kg m⁻¹)

For a string fixed at both ends, standing waves form when an integer number of half‑wavelengths fit into the length L:

L = n λ/2 (n = 1, 2, 3, …)

Combining with v = f λ gives the resonant frequencies:

f_n = n v / (2L) = n/(2L) √(T/μ)

5. Vibration of String and Overtones

The fundamental frequency (first harmonic) of a fixed‑fixed string is:

f₁ = v / (2L) = (1/2L) √(T/μ)

Higher harmonics are integer multiples of the fundamental:

First overtone (second harmonic): f₂ = 2f₁
Second overtone (third harmonic): f₃ = 3f₁
Third overtone (fourth harmonic): f₄ = 4f₁

Thus, unlike a closed pipe, a string supports all harmonics.

Standing wave patterns on a string fixed at both ends showing fundamental and overtones — Figure 3: Standing wave patterns on a string fixed at both ends (fundamental, second, and third harmonics).

6. Laws of Vibration of a Fixed String

Three empirical laws describe how the fundamental frequency varies with length, tension, and linear density while keeping the other two factors constant.

Law of Length (constant T, μ):
f ∝ 1/L → Doubling the length halves the frequency.
Law of Tension (constant L, μ):
f ∝ √T → Quadrupling the tension doubles the frequency.
Law of Linear Density (constant L, T):
f ∝ 1/√μ → Using a string four times as dense reduces the frequency by half.

These laws can be derived directly from the fundamental frequency expression:

f₁ = (1/2L) √(T/μ)

which shows the explicit dependence on each variable.

Worked Examples

Example 1: Closed Pipe Frequency

A closed pipe of length 0.50 m and radius 1.5 cm is filled with air (speed of sound v ≈ 340 m s⁻¹). Calculate the fundamental frequency, including end correction.

End correction: e = 0.61 r = 0.61 × 0.015 m = 0.00915 m
Effective length: L_eff = L + e = 0.50 m + 0.00915 m = 0.50915 m
Fundamental: f = v / (4 L_eff) = 340 / (4 × 0.50915) ≈ 166.8 Hz

Example 2: String Tension for a Desired Note

A guitar string of length 0.65 m and linear density 5.0 × 10⁻⁴ kg m⁻¹ is to produce a fundamental frequency of 330 Hz (E₄). Find the required tension.

From f₁ = (1/2L) √(T/μ) solve for T:
T = (2L f₁)² μ
T = (2 × 0.65 m × 330 Hz)² × 5.0×10⁻⁴ kg m⁻¹
T = (429)² × 5.0×10⁻⁴ ≈ 184 041 × 5.0×10⁻⁴ ≈ 92 N

Thus, a tension of about 92 N is needed.

Summary

This chapter has covered the essential physics of stationary waves in pipes and strings:

Boundary conditions dictate node/antinode placement, leading to distinct harmonic series for closed and open pipes.
End correction adjusts the effective length of pipes, improving the accuracy of frequency predictions.
Transverse wave speed on a string depends on tension and linear density, giving rise to a harmonic series that includes all integer multiples.
The three laws of string vibration (length, tension, density) provide quick scalars for instrument design and tuning.

Understanding these principles is crucial for analyzing musical instruments, acoustics, and various engineering applications involving wave phenomena.