Physics

Wave and Sound - 4 Session

Session Objectives

Session Objective Beats Conditions under which beats occur Standing Waves in air columns - All Conditions Standing waves in a string (with all conditions) Fundamental Frequency

Standing waves in a string (with all conditions) Let the waves with same amplitude, same frequency, having a phase difference ‘  ’, superimpose to produce a standing wave given by

String fixed at both ends A NN L x = L x = O Since x = 0 and x = L are nodes, at these points y = 0 for all ‘t’. Hence, for x = 0, Considering  = 0,

String fixed at both ends Similarly, for x = L where n = 0, 1, 2, …

String fixed at both ends If the length of the string is an integral multiple of, standing waves are produced. Hence, the other natural frequencies with which standing waves can be formed on string are Now

String fixed at both ends [First overtone or second harmonic] [Second overtone or third harmonic] [Third overtone or fourth harmonic]

String fixed at both ends N N AA N N N A A N N Second overtone Third harmonic First overtone Second harmonic

String fixed at one end N A A N A N N A A N N A Fundamental First overtone Second overtone

String fixed at one end The boundary condition that x = 0 is automatically satisfied by y = 2A sinkx cost. For x = L to be an antinode,

String fixed at one end Putting n = 0, [Fundamental frequency] Similarly, [First overtone] [Second overtone]

Fundamental Frequency The lowest frequency with which a standing wave can be set up in a string is called the fundamental frequency.

Standing Waves in Air Columns L Fundamental Frequency or First Harmonic L First Overtone or Third Harmonic L Second Overtone or Fifth Harmonic

Standing Waves in Air Columns L Fundamental Frequency or First Harmonic L First Overtone or Second Harmonic L Second Overtone or Third Harmonic

Beats The difference between two combining frequencies is called beats. Beats are produced due to superposition of two or more waves having nearly equal frequencies.

Class Test

Class Exercise - 1 An open organ pipe of length L vibrates in its fundamental mode. The pressure variation is maximum (a) at the two ends (b) at the middle of the pipe (c) at distance inside the ends (d) at distance inside the ends

Solution Hence answer is (a). Pressure variation is maximum at the ends.

Class Exercise - 2 An open pipe is suddenly closed at one end with the result that the frequency of third harmonic of the closed pipe is found to be higher by 100 Hz than fundamental frequency of the open pipe. The fundamental frequency of the open pipe is (a) 200 Hz(b) 300 Hz (c) 240 Hz(d) 480 Hz

Solution Hence answer is (a).

Class Exercise - 3 A pipe, open at both ends, gives frequencies which are (a) only even multiple of fundamental frequency (b) only odd multiple of fundamental frequency (c) all integral multiple of fundamental frequency (d) all fractional multiple of fundamental frequency

Solution Hence answer is (c). Open pipe produces all integral multiple of fundamental frequency.

Class Exercise - 4 In a closed organ pipe, the fundamental frequency is ‘’. What will be the ratio of the frequencies of the next three overtones? (a) 2 : 3 : 4(b) 3 : 4 : 5 (c) 3 : 7 : 11(d) 3 : 5 : 7

Solution Hence answer is (d). Closed pipe produces only odd multiples of fundamental frequency.

Class Exercise - 5 Two open organ pipes of length 25 cm and 25.5 cm produce 0.1 beats per second. The velocity of sound is (a) 350 m/s(b) m/s (c) 303 m/s(d) 255 m/s

Solution Hence answer is (d).  v = 255 m/s

Class Exercise - 6 Velocity of sound in air is 320 m/s. A pipe closed at one end has a length of 1 m. Neglecting end correction, the air column in the pipe can resonate for sound of frequency which is equal to (More than one options may be correct) (a) 80 Hz(b) 240 Hz (c) 320 Hz(d) 400 Hz

Solution Hence answer is (a, b, d). Fundamental frequency Hence, possible frequencies are odd multiples of fundamental frequency.

Class Exercise - 7 Two waves of wavelengths 2 m and 2.02 m respectively, moving with the same velocity superimpose to produce 2 beats per second. The velocity of the wave is equal to (a) 400 m/s(b) 402 m/s (c) 404 m/s(d) 406 m/s

Solution Hence answer is (c).  v = 404 m/s

Class Exercise - 8 A sonometer wire, 65 cm long, is in resonance with a tuning fork of frequency N. If the length of the wire is decreased by 1 cm and it is vibrated with the same tuning fork, 8 beats are heard per second. What is the value of N? (a) 256 Hz(b) 384 Hz (c) 480 Hz(d) 512 Hz

Solution Hence answer is (a). Given Also or Hence, N is calculated.

Class Exercise - 9 Beats are result of (a) diffraction (b) destructive interference (c) constructive interference (d) superposition of two waves of nearly equal frequencies

Solution Hence answer is (d). Definition of beats.

Class Exercise - 10 An organ pipe vibrates in fundamental resonance with the medium as air, nitrogen and oxygen. Which is the correct statement? (a) The wavelength changes with medium change (b) Both wavelength and frequency change (c) Both and remain unaltered with the medium change (d) The frequency changes with the medium change

Solution Hence answer is (a). Only wavelength depends on the medium.

Thank you