4.5 Standing Waves

A guitar string is stretched and fixed at each end so that the length is 75 cm. It is found to vibrate with resonant frequencies at 240 Hz and 360 Hz. There are no other resonant points between these two.

  What is the lowest resonant frequency which can oscillate on the guitar string?

A copper pipe of diameter 15 mm and length 45 cm is sealed closed at one end. A string is attached such that the copper pipe can be quickly spun around, causing the effect of a wind blowing across the open end at different frequencies.

What is the wavelength of the third harmonic?

A standing wave on a string between two fixed points sounds its second harmonic at frequency f.

Which expression, where n is an integer, gives the frequencies of harmonics that have an antinode at the centre of the string?

A pipe of length L₁ has one open end and one closed end. A second pipe, of length L₂  is open at both ends. The frequency of the first harmonic of both pipes is the same. 

What is the ratio ?

A flute is a musical instrument with a typical length, L, of 67 cm. It has a hole in the lip plate at one end which the musician blows across, and is open at the other end. 

The length of the air column is varied by opening and closing a series of keys along the side. The velocity of the air entering the lip plate is controlled by the musician.

To create a frequency f  the musician creates a wave with velocity v, holding down the keys such that the column of air is half the available length.

If she wants to increase the frequency to three times the original pitch, which changes to would achieve this?

	v	L
A.	× 9	× 2
B.	× 3	×
C.	stays the same	× 3
D.	×	×

A thin wire is held at tension with length L so that the speed of the waves on the string is equal to the speed of sound in air. At this tension the fundamental frequency of oscillation produces resonance in a pipe which is closed at one end.

Which of the pipes shown resonates?

A very long, empty measuring cylinder of cross sectional area 2.5 × 10⁻³ m² is set up so that water can be slowly run into it through a narrow, flexible tube.

A tuning fork is made to oscillate and held over the open end. A resonant frequency is heard early on in the experiment and then again after a further 5.5 × 10⁻³ m³ of water has run into the cylinder.

What is the wavelength of the wave produced by the tuning fork?

A string is stretched to a length where  L = 300 cm. At this length and tension it has resonant frequencies at 360 Hz and subsequently at 420 Hz.

What is the speed of the standing waves on the string?

Two microwave transmitters are set up to emit microwaves so that a stationary wave is created between them. The microwaves have a frequency of 300 MHz.

Which line correctly gives both the formula which represents the the distance between two adjacent nodes in the standing wave and the value for the distance?

	Formula	Distance
A.		50 cm
B.		2.0 m
C.		5.0 m
D.		20 cm

Which one of the following options is always true regarding the energy transferred along a standing wave and its amplitude?

An open pipe has sound waves sent through it causing it to have a standing wave of the third harmonic occurring within it. What is the nature of oscillations at the centre of the pipe (point X) and at its edge (point Y)?

The fundamental frequency for a pipe which is closed at one end, but open at the other, is 380 Hz. What is the frequency of its third harmonic?

An open pipe has both ends clear and has a standing wave formed within it. Which of the following statements is always true for this standing wave?

A wave of a certain frequency is setup on a string with two fixed ends which act as nodes

If the frequency is doubled, but nothing else is changed, how many nodes will be on the string?

Organs such as those found in churches often have many different kinds of pipes with both open and closed ends. For a single pipe that has a specific length, what is the ratio of the fundamental frequency with both ends open to the fundamental frequency with one end closed?

A pipe with a single closed end has a standing wave set up within it. If the length of the pipe is 2 m, which value below is closest to the wavelength of the seventh harmonic?

A standing sound wave is set up in an open pipe (both ends open) and the frequency of the first harmonic is 400 Hz. What is the approximate length of this pipe? (Assume the speed of sound is 343 m/s).

A pipe of length L is open at one end and closed at the other end of the pipe. What is the frequency of the fifth harmonic? (Assume the speed of sound in air is 343 m s^–1).