This document discusses standing waves on strings. It explains that when a string with fixed ends is plucked, waves travel back and forth along the string creating standing waves. The allowed wavelengths are λ=2L/m, where L is the string length and m is a positive integer. These standing waves are called the normal modes of vibration, and their frequencies are given by fm=mf1, where f1 is the fundamental frequency. The document provides an example problem involving calculating the wave speed, number of reflections, wavelengths, frequencies, and node spacing for the first two normal modes of a plucked guitar string.

1. Standing Waves on
Strings
BY ALYKHAN DEVSI

2. Introduction to standing waves on
strings
 Consider a string of length L with both fixed ends.
 When the string is plucked, waves travel back and forth along the string
creating standing waves.
 At the two ends of the string the amplitude is 0 and thus we have:
sin(
2𝜋
𝜆
𝐿) =0
2𝜋𝐿
𝜆
= 𝑚𝜋 where m is a positive, non zero integer
As a result we find that a string with both ends fixed can oscillate in a standing wave with
the following wavelengths:
𝜆 =
2𝐿
𝑚
, m = 1,2,3,4 …

3. Normal Modes
 Now that we have derived the equation for wavelengths of
standing waves on strings we see that the longest possible
wavelength is: 𝜆 =
2𝐿
1
= 2L and as the value of m increases the
wavelength decreases.
 These standing waves are called the normal modes of vibration,
and the frequencies corresponding to them are:
𝑓𝑚 =
𝑚
2𝐿
*
𝑇
𝜇
where T=tension and µ= mass per unit length
Because only m is changing we can write the relationship:
𝑓𝑚=m𝑓1, where 𝑓1 = 𝑓𝑢𝑛𝑑𝑎𝑚𝑒𝑛𝑡𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦

4. Harmonics or Resonant
Frequencies
 The allowed frequencies are called harmonics or resonant
frequencies, below are examples of the first 4:

5. Time for Some Practice!
 A guitar string is 0.50m long and has a linear mass density of
3.43e-4 kg/m. The tension in the string is kept at 85.0N.
 A) What is the wave speed in the string?
 B) If the string is plucked, approximately how many times would a wave
reflect from one end of the string in 2 seconds.
 C) What are the wavelengths and the frequencies of the first 2 normal
modes of vibration of the string.
 D) What is the spacing between two consecutive nodes for the first two
modes.

6. Solutions
 A)
The velocity depends on both the tension and the linear mass density:
𝑣 =
𝑇
𝜇
=
85.0
3.43𝑒−4
= 𝟒𝟗𝟕. 𝟖 𝐦/𝐬
 B)
The time is takes for the waves to travel from one end of the string to
the other is:
𝑡 =
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠𝑡𝑟𝑖𝑛𝑔
𝑤𝑎𝑣𝑒 𝑠𝑝𝑒𝑒𝑑
=
0.50
497.8
= 1.004𝑒 − 3 𝑠
Since reflection from a given end requires a back and forth trip, then
number of times a wave reflected from a given end in 2 seconds is :
2
2𝑡
=
2
2∗1.004𝑒−3
= 𝟗𝟗𝟓. 𝟔 𝒕𝒊𝒎𝒆𝒔

7. Solutions
 C)
For the first harmonic: 𝜆 =
2𝐿
1
= 2 ∗ 0.50 = 𝟏 𝒎
𝑓𝑚 =
𝑚
2𝐿
*
𝑇
𝜇
=
85.0
3.43𝑒−4
= 𝟒𝟗𝟕. 𝟖 𝑯𝒛
For the second harmonic: 𝜆 =
2𝐿
2
= 𝟎. 𝟓𝟎 m
𝑓𝑚= m𝑓1 = 2 ∗ 497.8 = 𝟗𝟗𝟓. 𝟔 𝑯𝒛

8. Solutions
 D)
The distance between two consecutive nodes is
𝜆
2
therefore
For the first harmonic:
D =
𝜆
2
=
2𝐿
2
= 𝟎. 𝟓 𝒎
For the second harmonic:
D =
𝜆
2
=
𝐿
2
= 𝟎. 𝟐𝟓 𝒎

9. Thank You!