Class 11 >> Physics >> oscillations-and-waves >> standing-waves-and-harmonics >> define-harmonics-in-pipes
Skip to Practice

define-harmonics-in-pipes

🚀 Harmonics in pipes refer to the specific frequencies at which a pipe can resonate. When a sound wave travels through a pipe, it can create standing waves, which are waves that remain in a constant position. These standing waves are formed by the interference of two waves traveling in opposite directions. The harmonics of a pipe depend on whether the pipe is open at both ends or closed at one end. In an open pipe, the harmonics are integer multiples of the fundamental frequency, while in a closed pipe, only odd harmonics are present.

Theory Explanation

Understanding Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. This results in points of no displacement called nodes and points of maximum displacement called antinodes.

Fundamental Frequency

The fundamental frequency (first harmonic) is the lowest frequency at which a pipe can resonate. For an open pipe, it is given by the formula f1 = v/2L, where v is the speed of sound in air and L is the length of the pipe.

\[ f_1 = \frac{v}{2L} \]
Harmonics in Open Pipes

In an open pipe, the harmonics are given by f_n = n(v/2L), where n is the harmonic number (1, 2, 3,...). This means that the frequencies are integer multiples of the fundamental frequency.

\[ f_n = n \cdot f_1 \]
Harmonics in Closed Pipes

In a closed pipe, only odd harmonics are present. The frequencies are given by f_n = n(v/4L), where n is an odd integer (1, 3, 5,...).

\[ f_n = n \cdot f_1 \]

Key Points

  • 🎯 Harmonics are specific frequencies at which a pipe resonates.
  • 🎯 Open pipes support all harmonics, while closed pipes support only odd harmonics.
  • 🎯 The fundamental frequency is the lowest frequency of vibration in the pipe.

🛠 Simulation is being generated. Please check back in a few moments.

Examples:💡

Calculate the fundamental frequency of a 2-meter long open pipe if the speed of sound is 340 m/s.

Solution:

Step 1: Use the formula for the fundamental frequency: f1 = v/2L.

\[ f_1 = \frac{340}{2 \times 2} = 85 \text{ Hz} \]

Find the first three harmonics of a closed pipe that is 1.5 meters long with the speed of sound as 340 m/s.

Solution:

Step 1: Calculate the fundamental frequency: f1 = v/4L = 340/(4*1.5) = 56.67 Hz.

\[ f_1 = \frac{340}{4 \times 1.5} = 56.67 \text{ Hz} \]

Step 2: The first harmonic (n=1) is 56.67 Hz, the second harmonic (n=3) is 170 Hz, and the third harmonic (n=5) is 283.33 Hz.

\[ f_3 = 3 \cdot f_1 = 3 \cdot 56.67 = 170 \text{ Hz} \]

Common Mistakes

  • Mistake: Confusing the harmonics of open and closed pipes.

    Correction: Remember that open pipes have all harmonics, while closed pipes only have odd harmonics.

  • Mistake: Incorrectly applying the formulas for frequency calculations.

    Correction: Always check whether the pipe is open or closed before using the formulas.

← Back Start Practice