Sound Wavesand Guitars

Sound waves are produced by vibrating objects, such as a guitar string.

INTRODUCTION

Sound is a longitudinal wave. As a guitar string vibrates, it causes surrounding air molecules to oscillate in a direction which is either parallel or anti-parallel to the propagation of the wave.

The displacement of the air particles results in a pressure wave composed of compressions (high pressure) and rarefractions (low pressure).

A sound wave is the alternating pattern of compressions and rarefractions.

SPEED OF SOUND

The speed of sound depends on the property of the medium though which it is propagating.

Calculating speed requires the introductions of bulk modules (B) which represents the ‘stiffness’ of air.

B= -V ∆p ∆V Using the bulk module and the density of the medium

(air), velocity of sound can be determined. v=

SPEED OF SOUND

Question 1

Calculate the speed of sound in air. (B= 1.01x105 Pa, ρ= 1.2 kg/m2)

SPEED OF SOUND

Answer: 290 m/s (NOTE: this answer is obtained from using the information provided in the textbook. The experimentally determined velocity of sound in air is 343 m/s)

Using the formula on the previous, divide B by the density and take the square root.

The decibel (dB) is a commonly used measurement of sound intensity or power.

  

DECIBELS

Decibels are referenced to the quietest sound that a human ear can detect (Io= 10-12 W/m2).

The following equation is used to determine any other sound intensity level

β(I)=0 dB+10 log(I/Io)

DECIBELS

Question 2

Calculate the sound intensity level of a note plucked by an unknown string that has an intensity of 7.5x10-6 W/m2..

DECIBELS

Answer: 68.75 dB

Using the equation on the previous slide and I0= 10-12

dB= 10 log(7.5x10-6/10-12)

The strings of a guitar have their own unique frequencies.

FREQUENCY

String Note Frequency

1 E (high) 329.6

2 B 246.9

3 G 196.0

4 D 146.8

5 A 110.0

6 E (low) 82.4

FREQUENCY

The frequency of the oscillation of air particles is the same as the frequency of the vibration in the guitar string.

The following slide is a video of me plucking the low E string on my guitar. The sound wave of the note has the same frequency as the wave of the string which can be seen.

FREQUENCY

Question 3

Calculate the fundamental frequency of the same unknown string given the length of the string to equal 80 cm and the velocity to equal 400 m/s. (Hint: λ= 2L when determining the fundamental frequency)

FREQUENCY

Answer: 250 Hz

f= v/λλ= (80 cm×2)/ 100 m= 1.6 m

f= (400 m/s)/(1.6 m)= 250 Hz which approximately corresponds to a B-string.

The phase relationship between the pressure variations and the displacement is as follows 

∆p=Bksmsin(kx-ωt+ϕ)B= Bulk modulusk= 2π/λω= 2π/Tsm= displacement

PRESSURE VARIATIONS

To find the amplitude of the pressure variations as the coefficient of the sine factor use the following equation

∆pm= Bksm

PRESSURE VARIATIONS

Question 4

Calculate the displacement amplitude of the sound waves of the B-string with a pressure amplitude of 5.0×10-2 Pa. (Use 343 m/s for velocity of sound in air)

PRESSURE VARIATIONS

Answer: 1.08×10-7 m

Known values: 343 m/s, 250 Hz, 1.01×105 Pa, 5.0×10-2 PaSm=∆p/Bk Sm= (5.0×10-2 Pa)/((1.01×105 Pa)(2π(250 Hz)/(343 m/s))

Thank you for watching!

Works Cited

Hawkes Et Al. Physics for Scientists and Engineers: An Interactive Approach. Vol. 1. Vancouver: U of British Columbia, n.d. Print.

"Standard Guitar Tuning – EADGBE | Pro Guitar Tuner." Pro Guitar Tuner. N.p., n.d. Web. 20 Feb. 2015.