Wave in Pipes and Strings

SYLLABUS

Stationary waves in closed and open pipes, Harmonics and overtones in closed and open organ pipes, End correction in pipes, Velocity of transverse waves along a stretched string, Vibration of string and overtones, Laws of vibration of fixed string.

What is This Chapter About?

Why does a flute produce a different sound from a trumpet? Why does a thick guitar string produce a lower note than a thin one? Why do organ pipes of different lengths play different musical notes? The answers all lie in the physics of standing waves — and specifically in how stationary waves are formed in confined systems like pipes and stretched strings.

Chapter 8 of NEB Grade 12 Physics explores one of the most musically and acoustically rich topics in the course. It shows how the standing wave patterns studied abstractly in Chapter 6 manifest concretely in musical instruments, and derives the precise mathematical relationships between the physical properties of pipes and strings and the frequencies they produce. Carrying 4 teaching hours, this chapter is compact but extremely important for both NEB examinations and for understanding the physics of sound and music.

Key Concepts You Will Learn

Stationary Waves in Pipes

When sound waves travel through a pipe, they reflect from the ends and superpose with the incoming waves to form stationary waves. The nature of these waves — and the frequencies produced — depends on whether the pipe ends are open or closed.

At a closed end, the air molecules cannot move freely, so a node (zero displacement) always forms. At an open end, the air molecules can move freely, so an antinode (maximum displacement) always forms. These boundary conditions determine which frequencies can form standing waves in a given pipe.

Closed Organ Pipes

In a closed pipe (closed at one end, open at the other), a node forms at the closed end and an antinode at the open end. The fundamental mode has a quarter wavelength fitting in the pipe length: [latex]l = \frac{\lambda}{4}[/latex], giving fundamental frequency [latex]f_1 = \frac{v}{4l}[/latex]. Only odd harmonics are present in a closed pipe — [latex]f_1, 3f_1, 5f_1, \ldots[/latex]. This is why closed pipes have a more muffled, hollow sound compared to open pipes.

Open Organ Pipes

In an open pipe (open at both ends), antinodes form at both ends. The fundamental mode has half a wavelength fitting in the pipe: [latex]l = \frac{\lambda}{2}[/latex], giving fundamental frequency [latex]f_1 = \frac{v}{2l}[/latex]. Both odd and even harmonics are present — [latex]f_1, 2f_1, 3f_1, \ldots[/latex] — giving open pipes a richer, brighter tone.

For the same length, an open pipe produces a fundamental frequency exactly twice that of a closed pipe. This is why open flutes sound higher than closed flutes of the same length.

End Correction

In practice, the antinode at an open end does not form exactly at the end of the pipe but slightly beyond it. This end correction (e) is approximately [latex]e = 0.6r[/latex], where r is the radius of the pipe. The effective length of an open pipe is [latex]l + 2e[/latex] and of a closed pipe is [latex]l + e[/latex]. End correction explains why pipes need to be slightly shorter than theory predicts to produce the desired pitch.

Velocity of Transverse Waves on a Stretched String

When a string is stretched under tension T and has a linear mass density [latex]\mu[/latex] (mass per unit length), transverse waves travel along it with speed: [latex]v = \sqrt{\frac{T}{\mu}}[/latex]. A tighter string (higher T) gives higher wave speed; a heavier string (higher [latex]\mu[/latex]) gives lower wave speed. This result directly explains why tightening a guitar string raises its pitch.

Vibration of a Stretched String

When a stretched string fixed at both ends is plucked or bowed, nodes form at both fixed ends. The fundamental mode has half a wavelength fitting between the ends: [latex]l = \frac{\lambda}{2}[/latex], giving: [latex]f_1 = \frac{1}{2l}\sqrt{\frac{T}{\mu}}[/latex]. Higher harmonics (overtones) are whole-number multiples of the fundamental: [latex]f_n = nf_1[/latex].

Laws of Vibration of a Fixed String

Three important laws govern string vibration, all derivable from [latex]f_1 = \frac{1}{2l}\sqrt{\frac{T}{\mu}}[/latex]:

The Law of Length states that frequency is inversely proportional to length at constant tension and linear density — halving the string length doubles the frequency. This is why pressing a guitar string against a fret shortens the vibrating length and raises the pitch.

The Law of Tension states that frequency is proportional to the square root of tension — quadrupling the tension doubles the frequency. This is why tightening a string raises its pitch.

The Law of Mass states that frequency is inversely proportional to the square root of linear mass density — a heavier string vibrates more slowly and produces a lower note. This is why the bass strings on a guitar are thicker.

Why This Chapter Matters for NEB Exams

Waves in Pipes and Strings is examined extensively in NEB Grade 12 Physics. Frequency of harmonics in open and closed pipes, derivation of speed of transverse waves on a string, and the laws of vibration of a fixed string are all standard examination topics. Numerical problems — finding the fundamental frequency of a pipe or string given its length, tension, and density — appear in virtually every board paper. End correction is a frequently asked short-answer question.

Important Topics to Focus On

• Formation of stationary waves in closed and open pipes — boundary conditions
• Fundamental frequency and harmonics of a closed pipe: [latex]f_n = \frac{(2n-1)v}{4l}[/latex]
• Fundamental frequency and harmonics of an open pipe: [latex]f_n = \frac{nv}{2l}[/latex]
• Comparison of closed and open pipes — why open pipes have richer tone
• End correction in pipes: [latex]e = 0.6r[/latex]
• Derivation of speed of transverse wave: [latex]v = \sqrt{\frac{T}{\mu}}[/latex]
• Fundamental frequency of a stretched string: [latex]f_1 = \frac{1}{2l}\sqrt{\frac{T}{\mu}}[/latex]
• Laws of vibration of a fixed string — length, tension, and mass laws
• Numerical problems on pipe frequencies, string frequencies, and wave speed

Connection to Other Chapters

Waves in Pipes and Strings builds directly on Chapter 6 (Wave Motion) — stationary waves in pipes and strings are the physical realisation of the mathematical standing wave patterns introduced there. Chapter 7 (Mechanical Waves) provides the speed of sound used in pipe frequency calculations. Chapter 9 (Acoustic Phenomena) extends these ideas to the characteristics of musical sound — intensity, loudness, pitch, and quality are all directly related to the harmonic content of vibrating strings and pipes. The speed of transverse waves on a string also connects to Chapter 7 of Grade 11 (Gravitation) through dimensional analysis — a classic examination problem.