Figure 1: Resonance – The First 3 Harmonics

Lab: AN EXAMINATION OF STANDING WAVES ON A STRING

Theory If the two ends of a string are fixed and some point on the string is set into transverse oscillation with

frequency f, the waves propagate to both ends of the string where they reflect and travel back in the other

direction. Under certain conditions, these two waves travelling in opposite directions interfere in such a

way as to produce standing waves in the string. In the standing waves there are positions called nodes

where the string does not move and places in between called antinodes or loops where the string motion is

a maximum. Since both ends of the string are fixed they must be nodes; therefore, the condition under

which standing waves are produced is that the length of the string L, is an integer number of half-

wavelengths:

(1) L n

2

In this experiment you will vibrate a string

using a ticker-timer at a fixed frequency of

60 1 Hz (the frequency of A.C. supplied

by the power company).

The speed of the wave along a string depends

only on the tension, T in the string and the

mass per unit length of the string (the linear

density). The speed of the wave is given by

(2) vT

For any wave the velocity is the product of

the frequency, f and the wavelength

(3) v f

Equations (2) and (3) can be combined and v eliminated to obtain

(4) fT

1

THE EXPERIMENT Purpose

The purpose of the experiment is to verify the above theory for waves on a string.

12

2

Apparatus

string, dial-o-gram or electronic balance, triple beam balance, ticker-timer (used as a 60 Hz wave

generator), meter stick, slotted mass holder (or paper clip), slotted masses (or metal washers), pulley and

clamp, c-clamp

Method

Part A: Determination of the Linear Density of a String by Direct Measurement

Determine the linear density of the string using a dial-o-gram or electronic balance and a length of the

string supplied by your teacher. Use a piece of string that is just under 1.000 m in length and measure its

mass to the nearest hundredth of a gram ( kg 00001.0 ). Record your observations and l. density below.

mass (m): _________ kg, length (L): _________ m, linear density of string (), _________ kg/m

Part B: Determination of the Linear Density of a String by Standing Wave Analysis

Cut a length of the string that is 1.5 to 2.0 m long so it is longer than the length of your lab table. Attach

one end to the hammer of the ticker-timer using the wing-nut (please raise the hammer first so it will not

hit the striker plate). Run the other end of the string over a pulley clamped to the one end of your table.

Lightly c-clamp the ticker-timer to the other end of the table.

Tie a loop at the end of the string that passes over the pulley so that you can attach a weight holder. You

can hook a straightened paperclip to the loop to act as the weight holder. Increase the tension in the string

(T = mg) by attaching mass (washers, paperclips, 50 g and 100 g slotted masses, etc.) to the weight

holder.

Plug in and turn on the ticker-timer. Adjust the tension in the string by adding or subtracting mass until

standing waves are seen with well-defined fixed nodes and maximum amplitude. By changing the tension,

the string can be made to resonate with two or more nodes. The distance between the nodes is one-half the

wavelength.

Figure II: The Fourth Harmonic (n=4), L 42

Organize your observations and calculations in a table like the one below.

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Table I: Standing Waves on a String (obtain data for at least three harmonics)

number of

antinodes

(harmonic

number, n)

distance

between

nodes (m)

wavelength

(m)

mass (kg)

(include the

holder)

tension

(hanging

weight)

(N)

wave speed

(m/s)

linear

density ()

of string

(kg/m)

6

5

4

3

2

1 Its best if you omit this one as the large weight/tension is hard on the wave generator.

Count and record the number of antinodes on the string. Begin with five or six antinodes then reduce the

number of antinodes by adding weight in subsequent trials. Measure and record the distance between any

adjacent pair of nodes. Determine and record the wavelength. Refer to equation (1). Measure the mass

hanging from the string including the holder using a triple beam balance. Record the mass tensioning the

string in your data table. Calculate and record the tension in the string for each harmonic (at least three).

Calculate and record the speed of the wave in each case using equation (3). Finally, obtain an expression

for the linear density () of the string using equation (4). Use this formula to calculate the linear density

of the string for each case. Calculate the average linear density of the string. Be respectful of the rules for

significant figures. Calculate the average relative error.

CONCLUSION/DISCUSSION QUESTIONS (Please do the following work on loose-leaf paper.)

1. Discuss and compare your result in part A with part B. (2 marks)

2. In a general way:

a) how does the number of antinodes relate to the wavelength

b) how does the tension affect the number of antinodes on the resonant string?

c) how is the wavelength related to the tension?

d) how is the speed related to the tension?

e) how is the wavelength related to the wave’s speed?

3. Show that equation (2) is dimensionally correct; that is the units on the right side equal to the units on

the left.

4. Assume we could change the frequency of the wave generator used in this lab and the tension is fixed.

a) How would a doubling of the frequency affect the wavelength?

b) A standing wave is set-up on a string. The resonant mode is the fourth harmonic as shown in

figure II above. Explain what would happen to the standing wave pattern if the frequency of the

wave generator was doubled say from 60 Hz to 120 Hz. Sketch what you would see. What would

happen if the frequency was tripled? What would happen if the frequency was halved (f = 30 Hz)?

The following questions are not directly related to this lab.

5. In the previous speed of sound lab we used a fixed frequency source (tuning fork) to produce standing

waves by varying the length of the air column. In this lab, the medium length (string length) was fixed

and the source frequency was fixed. How were you able to produce standing waves? What did you

change?

6. Suppose you want to produce standing waves in an air column but had a fixed length air column and a

fixed frequency source, how might you be able to make standing waves? Explain.