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Standing Waves on a String

Casey Robinson

Robert Bianchini

29 January 2009

Version 1

Abstract

This experiment is designed to investigate the properties of standing waves on a

string. There are many properties to investigate; propagation velocity and vibration

modes are two that come to mind. The equation for propagation velocity/frequency was

verified, and the relationship between vibration modes was confirmed.

Introduction

When a string stretched between two fixed endpoints is perturbed a wave

travels across the string. When this propagating wave reaches an endpoint it is

reflected and returns inverted. If another wave is coming towards the reflected wave,

interference will occur. Interference can be constructive or destructive. Constructive

creates a larger wave and destructive creates a smaller wave.

Any wave propagating across a string can be described as a combination of the

string’s fundamental frequency and its harmonic frequencies. Interference of identical

waves at one of these frequencies will result in a standing wave on the string.

Standing waves have many unique characteristics. Standing waves are static

across the length of the string and only oscillate perpendicular to the string. Nodes,

points where the string does not move, are formed at even intervals. Anti-nodes, the

points at which the string oscillations have the largest amplitude, occur exactly in

between each of the nodes.

Theoretical

Waves on a string follow the Wave Equation (1), which can be found in any introductory

physics text. In the wave equation c is the speed and u(x,t) is the traveling wave.

𝜕^2 𝑢

𝜕𝑡^2

= 𝑐^2

𝜕^2 𝑢

𝜕𝑥^2 (1)

Figure 1: Experimental Setup mass

driver detector

string

clamp

Experimental Method

A Pasco Sonometer (WA-9611/9613) was setup to facilitate our investigation

into the properties of standing waves on a string. The sonometer consists of four parts;

the clamp which holds the string in place, the bridge supports which designate the

endpoints of the string, the tensioning lever which provides a location to attach mass

for tension, and the base which holds it all together.

The sonometer was placed on a workbench in the lab. Four guitar strings of

different linear density were used in this experiment to investigate the effect of

linear density on propagation velocity. See table 1 for further details. Varying masses

were attached to the tensioning lever to apply various amounts of tension to the string.

These are discussed in table 5.

An audio driver was connected to a frequency generator through an audio

amplifier. The driver was placed underneath the string 5cm to the right of the left

bridge support. The electric field produced from the driver forced the guitar string to

oscillate. This oscillation was captured by the detector, placed 5cm to the left of the

right bridge support. Both the frequency generator and the detector were displayed on

an oscilloscope.

This lab tries to answer many questions about standing waves and each one

requires a different variation in the setup of the experiment.

Does the location of the nodes and antinodes depend on the tension and linear

density? What is the mathematical relationship between propagation velocity, tension,

and linear density? To answer these questions the sonometer was set up to have a

string length of 50cm with the driver and detector 5cm from opposite endpoints.

Masses ranging from 200g to 1kg were placed on the fifth notch of the tensioning

lever, resulting in tensions from 9.81 N to 49.05 N. Adjusting the frequency of the

driver allowed for observation of standing waves on the string at each tension. After

standing waves at the fundamental frequency were found for each tension, the string

was swapped for one with smaller linear density. The experiment was repeated with

three different strings. The results of this portion of the experiment can be found in

table 5.

What is the mathematical relationship between the fundamental frequency and

the harmonic frequencies? In order to answer this question string one was stretched

across the sonometer and one kilogram was placed on the first notch of the tensioning

lever. The bridge supports were placed at three different distances; 50cm for setup

#1, 30cm for setup #2, and 40cm for setup #3. For each distance the driver and

detector were placed 5cm from opposite endpoints. This was also to prove that the

change in length did not affect the relationship between the frequencies. The results

of this portion of the experiment can be found in tables 2, 3, and 4.

String 1 String 2 String 3 200g 20.745 ± 0.001 Hz 34.872 ± 0.001 Hz 26.193 ± 0.001 Hz 400g 28.368 ± 0.001 Hz 48.384 ± 0.001 Hz 36.753 ± 0.001 Hz 600g 35.122 ± 0.001 Hz 59.328 ± 0.001 Hz 46.063 ± 0.01 Hz 800g 40.197 ± 0.01 Hz 68.267 ± 0.01 Hz 51.635 ± 0.01 Hz 1000g 44.403 ± 0.01 Hz 75.557 ± 0.01 Hz 58.495 ± 0.01 Hz String 1 String 2 String 3 200g 20.826 ± 0.06 Hz 35.061 ± 0.06 Hz 26.995 ± 0.06 Hz 400g 29.452 ± 0.06 Hz 49.584 ± 0.06 Hz 38.176 ± 0.06 Hz 600g 36.071 ± 0.06 Hz 60.727 ± 0.06 Hz 46.756 ± 0.06 Hz 800g 41.651 ± 0.06 Hz 70.122 ± 0.06 Hz 53.989 ± 0.06 Hz 1000g 46.568 ± 0.06 Hz 78.398 ± 0.06 Hz 60.362 ± 0.06 Hz

Analysis of data

Linear Density

Mass

𝐿𝑒𝑛𝑔𝑡 𝑕 =^

0.1736 g

3.07 𝑐𝑚 = 0.0565 g/cm

ΔMass 𝑀𝑎𝑠𝑠 2

ΔLength 𝐿𝑒𝑛𝑔𝑡 𝑕 2

.

2

2

Tension

T = (notch position)(mass)(gravity) = 5 * 200g * 9.

𝑚 𝑠^2

= 9.81 N

ΔT =

ΔMass 𝑀𝑎𝑠𝑠 2

ΔGravity 𝐺𝑟𝑎𝑣𝑖𝑡𝑦 2

Δ 1 200 2

.

2

Frequency

f =

𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑊𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡 𝑕

𝑇𝑒𝑛𝑠𝑖𝑜𝑛 𝐿𝑖𝑛𝑒𝑎𝑟 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 2 ∗𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒

9.81 𝑁 0.0565 𝑔/𝑐𝑚 2 ∗ 50 𝑐𝑚

=20.826 Hz

Δf =

ΔT 2 ∗𝑇 2

Δμ 2 ∗μ 2

ΔD 𝐷 2

. 2 ∗9. 2

. 2 ∗. 2

.

2

Table 5 : Experimental frequency dependence on density and applied mass Table 6 : Theoretical frequency dependence on density and applied mass

Uncertainty Evaluation

There are a few fundamental sources of error in this experiment. The limitation

of measurement devices is the most obvious one. Air resistance is not included in the

simple theoretical model described above, which of course exists in the lab. Also

magnetic force holding the bridge supports was not strong enough to create a perfect

fixed endpoint. At times, particularly with high frequency, the string tended to bounce

at the ends. This resulted in difficulty obtaining an accurate measurement of standing

wave frequencies.

To overcome the limitation of the bridge supports, mass was placed on each of

the supports. The gravitational force of the masses helped to fix the problem of

bouncing strings.

Discussion/Conclusion

This experiment shows that the mathematical relationship for standing wave

frequency derived above is accurate. The results of the experiment are not exact due

to some limitations in the setup. Also the relationship between the fundamental

frequency and the successive harmonic frequencies is correct.

If this experiment were to be repeated a few things could be different to

improve the accuracy of the results. Placing the sonometer into a vacuum would

eliminate air resistance, just as in the wave equation. Adding a stronger bonding force

between the bridge supports and the base would eliminate some of the bounce in the

string. This would make finding a frequency for standing waves more accurate.