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Stationary waves questions
Two waves of the same frequency travelling in opposite directions superpose into a pattern that stays put. Fix both ends of a string and only certain wavelengths fit, which gives the string a set of natural frequencies. The same argument applied to a column of air explains the notes a wind instrument can play.
19 original questions · 55 marks · the stationary waves notes · Waves
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Describe how a stationary wave is formed on a string.
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Two progressive waves of the same frequency and amplitude travel in opposite directions (for example an incident wave and its reflection) and superpose (1), producing a pattern of nodes and antinodes that does not move along the string (1).State the distance between two adjacent nodes on a stationary wave, in terms of the wavelength λ.
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Half a wavelength, λ/2 (1).Distinguish between a node and an antinode.
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A node is a point of zero amplitude (no displacement) (1); an antinode is a point of maximum amplitude (1).The frequency of the first harmonic of a stretched string is given by f1 = (1/2L)√(T/μ). State what the symbols T and μ represent, and give the SI unit of μ.
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T is the tension in the string (1). μ is the mass per unit length of the string, measured in kg m−1 (1).State why a stationary wave on a string fixed at both ends must have a node at each end.
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The ends are fixed and cannot move, so their displacement is always zero, which is the definition of a node (1).Points P and Q lie between the same pair of adjacent nodes on a stationary wave. Point R lies between the next pair of nodes along. State the phase difference between P and Q, and between P and R.
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P and Q: zero, they oscillate in phase (1). P and R: π rad (180°), they oscillate in antiphase (1).A string of length 0.60 m is fixed at both ends and vibrates in its fundamental mode. The wave speed on the string is 240 m s−1. Calculate (a) the wavelength and (b) the fundamental frequency.
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(a) Fundamental: λ = 2L = 2 × 0.60 = 1.2 m (1)
(b) f = v/λ = 240/1.2 (1)
f = 200 Hz (1)A string of length 0.60 m is fixed at both ends. The speed of waves on the string is 240 m s−1. Calculate the frequencies of the second and third harmonics.
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Second harmonic: λ = L = 0.60 m (1)
f = 240/0.60 = 400 Hz (1)
Third harmonic: f = 3v/2L = 600 Hz (1)A string of length 0.80 m fixed at both ends vibrates at its fundamental frequency of 150 Hz. Calculate the speed of the waves on the string.
A stationary wave has a wavelength of 0.50 m. Calculate the distance between adjacent nodes, and the distance between a node and the next antinode.
A guitar string has a vibrating length of 0.750 m, and this length of string has a mass of 1.20 g. The string is under a tension of 90.0 N. Calculate the frequency of the first harmonic. Use f1 = (1/2L)√(T/μ).
A string fixed at both ends vibrates at its first-harmonic frequency. The length of the string and the string itself are unchanged. Determine the factor by which the tension must change for the first-harmonic frequency to double.
A string of length 1.2 m fixed at both ends vibrates in its fourth harmonic. State the number of nodes and the number of antinodes on the string, and calculate the wavelength.
A string of length 1.2 m fixed at both ends carries waves travelling at 180 m s−1. Calculate the frequencies of the first three harmonics.
A string of length 1.0 m fixed at both ends vibrates at 300 Hz, and the wave speed on it is 200 m s−1. Determine which harmonic this is.
State three differences between a progressive wave and a stationary wave.
A violin string has a vibrating length of 0.328 m and a mass per unit length of 0.60 g m−1. The maker states that when the tension is 50.0 N the first harmonic of the string is concert pitch A, 440 Hz. Deduce whether the maker's statement is correct. Use f1 = (1/2L)√(T/μ).
The driving frequency applied to a stretched string of fixed length 1.5 m is increased slowly. Stationary waves form at 240 Hz and next at 300 Hz, with none in between. Determine the frequency of the first harmonic and the speed of the waves on the string.
Describe an experiment to investigate how the first-harmonic frequency of a stretched string depends on its vibrating length L. Your answer should include how the stationary wave is produced, the measurements taken, and how the results would be used to show that f1 is inversely proportional to L.
The same practice on paper: the printable workbook for this topic, questions and a worked answer book.
Practise stationary waves one question at a time
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