Waves and Sound (part 1)

Lecture 2

Outline

• Types of Waves: Transverse and Longitudinal• Periodic Waves• The Speed of Wave in a String• Producing a Sound Wave• Using a Tuning Fork to Produce a Sound Wave• Speed of Sound• Sound Intensity• Decibel

The Nature of Waves

1. A wave is a traveling disturbance.

2. A wave carries energy from place to place.

Transverse Wave Longitudinal Wave

Types of Waves – Transverse

• In a transverse wave, each element that is disturbed moves in a direction perpendicular to the wave motion

Types of Waves – Longitudinal

• In a longitudinal wave, the elements of the medium undergo displacements parallel to the motion of the wave

• A longitudinal wave is also called a compression wave

Other Types of Waves

• Waves may be a combination of transverse and longitudinal

• Water waves are partially transverse and partially longitudinal

Types of Waves – Traveling Waves

• Flip one end of a long rope that is under tension and fixed at the other end

• The pulse travels to the right with a definite speed

• A disturbance of this type is called a traveling wave

Periodic Waves

Periodic waves consist of cycles or patterns that are produced over and over again by the source.

In the figures, every segment of the slinky vibrates in simple harmonic motion, provided the end of the slinky is moved in simple harmonic motion.

Displacement – distance graph Displacement – time graph

The amplitude A is the maximum distance of a particle of the medium from the particles’ undisturbed position (equilibrium position).

The wavelength is the horizontal length of one cycle of the wave (The distance between two crests).

The period T is the time for one oscillation of the particle in the medium.

The frequency f is the number of oscillation per second of the particle in the medium.

Wavelength versus Period• The first graph is called the displacement – distance graph.• It shows the profile of the wave at one particular instant.• The second graph is called the displacement – time graph .• It shows how a particle at a particle location of the wave

varies its displacement with time.• Note that : The length between two crests for

displacement – time graph is λ, the wave length• The time interval between two crests for displacement – time graph is the period

Producing a continuos travelling wave in a string

• The left end of the string is connected to a blade that is set vibrating.

• Every part of the string, such as point P oscillates vertically with SHM

Velocity = Frequency x WavelengthAfter the particle vibrate once, the wave moves forward by a wavelength. i.e. one λ passes through point P in one T

1

Velocity,

fv

TTv

t

sv

AM and FM radio waves are transverse waves consisting of electric and magnetic field disturbances traveling at a speed of 3.00 108 m/s. A station broadcasts AM radio waves whose frequency is 1230 103 Hz and an FM radio wave whose frequency is 91.9 106 Hz. Find the distance between adjacent crests in each wave.(Ans : 244m , 3.26 m)

Example 1:

Solution:

A wave traveling in the positive x-direction is pictured in Figure. Find the amplitude, wavelength, speed, and period of the wave if it has a frequency of 8.00 Hz. In Figure, x = 40.0 cm and y = 15.0 cm.(Ans : 0.400 m, 0.150m , 3.20 m/s, 0.125s)

Example 2:

Solution:

Speed of a Wave on a String

• The speed on a wave stretched under some tension, F

– m is called the linear density, m and L are denoted as mass and length of the wire.

• The speed depends only upon the properties of the medium through which the disturbance travels

Transverse waves travel on each string of an electric guitar after the string is plucked. The length of each string between its two fixed ends is 0.628 m, and the mass is 0.208 g for the highest pitched E string and 3.32 g for the lowest pitched E string. Each string is under a tension of 226 N. Find the speeds of the waves on the two strings.(Ans : 826 m/s, 207 m/s)

Example 3:

Solution:

A uniform string has a mass M of 0.0300 kg and a length L of 6.00 m. Tension is maintained in the string by suspending a block of mass m = 2.00 kg from one end .(a) Find the speed of a transverse wave pulse on this string.(b) Find the time it takes the pulse to travel from the wall to the

pulley. Neglect the mass of the hanging part of the string.(Ans : (a) 62.6 m/s(b)0.0799s)

Example 4:

Solution:

Interference of Waves

• Two traveling waves can meet and pass through each other without being destroyed or even altered

• Waves obey the Superposition Principle– When two or more traveling waves encounter each

other while moving through a medium, the resulting wave is found by adding together the displacements of the individual waves point by point

– Actually only true for waves with small amplitudes

Constructive Interference

Destructive Interference

Reflection of Waves – Fixed End

• Whenever a traveling wave reaches a boundary, some or all of the wave is reflected

• When it is reflected from a fixed end, the wave is inverted

• The shape remains the same

Reflected Wave – Free End

• When a traveling wave reaches a boundary, all or part of it is reflected

• When reflected from a free end, the pulse is not inverted

Sound WavesSound waves are longitudinal waves

Producing a Sound Wave

• Any sound wave has its source in a vibrating object

• Sound waves are longitudinal waves (also known as mechanical or pressure wave because compressed and stretched) traveling through a medium

• A tuning fork can be used as an example of producing a sound wave

Using a Tuning Fork to Produce a Sound Wave

• A tuning fork will produce a pure musical note

• As the tines vibrate, they disturb the air near them

• As the tine swings to the right, it forces the air molecules near it closer together

• This produces a high density area in the air– This is an area of compression

Using a Tuning Fork, cont.

• As the tine moves toward the left, the air molecules to the right of the tine spread out

• This produces an area of low density– This area is called a

rarefaction

Using a Tuning Fork, final

• As the tuning fork continues to vibrate, a succession of compressions and rarefactions spread out from the fork

• A sinusoidal curve can be used to represent the longitudinal wave– Crests (maximum points) correspond to compressions and

troughs (minimum points) to rarefactions

Categories of Sound Waves

• Audible waves– Lay within the normal range of hearing of the human ear– Normally between 20 Hz to 20 000 Hz

• Infrasonic waves– Frequencies are below the audible range– Earthquakes are an example

• Ultrasonic waves– Frequencies are above the audible range– Dog whistles are an example

Speed of Sound, General

• The speed of sound is higher in solids than in gases– The molecules in a solid interact more strongly

• The speed is slower in liquids than in solids– Liquids are more compressible

(liquid) (solid)

Cont.

,

,

Cont.

The Speed of Sound

Sound travels through gases, liquids, and solids at considerably different speeds.

Speed of Sound in Air

• 331 m/s is the speed of sound at 0° C• T is the absolute temperature

or v = 331 + 0.6(T)where T in degree Celcius (°C)

A stone is dropped into a well. The splash is heard 3.00 s later. What is the depth of the well?Assume the temperature is at 20.0C.(Ans : 40.7m)

Example 5:

Solution:

Sound Intensity

Sound waves carry energy that can be used to do work.The amount of energy transported per second is called the power of the wave.The sound intensity is defined as the power that passes perpendicularly through a surface divided by the area of that surface.

SI unit: W/m2

𝐼 (𝑆𝑜𝑢𝑛𝑑 𝐼𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 )= 𝑃 (𝑃𝑜𝑤𝑒𝑟 )𝐴 ( 𝐴𝑟𝑒𝑎 )

12 x 10-5 W of sound power passed through the surfaces labeled 1 and 2. The areas of these surfaces are 4.0 m2 and 12 m2. Determine the sound intensity at each surface.(Ans: 3.0 x 10-5W/m2 , 1.0 x 10-5W/m2 )

Example 6:

Solution:

Intensity Level of Sound Waves

• The sensation of loudness is logarithmic in the human ear

• β is the intensity level or the decibel level of the sound

• is the threshold of hearing

Intensity vs. Intensity Level

• Intensity is a physical quantity• Intensity level is a convenient mathematical

transformation of intensity to a logarithmic scale

A noisy grinding machine in a factory produces a sound intensity of 1.00×1025 W/m2. Calculate (a) the decibel level of this machine and (b) the new intensity level when a second, identical

machine is added to the factory. (c) A certain number of additional such machines

are put into operation alongside these two machines. When all the machines are running at the same time the decibel level is 77.0 dB. Find the sound intensity.

(Ans : (a) 70.0 dB, 73.0 dB, 5.01 x 10-5W/m2 )

Example 7:

Solution:

Audio system 1 produces a sound intensity level of 90.0 dB, and system 2 produces an intensity level of 93.0 dB. Determine the ratio of intensities.(Ans : 2.00)

Example 8:

Solution:

Spherical Waves

• A spherical wave propagates radially outward from the oscillating sphere

• The energy propagates equally in all directions

• The intensity is

Intensity of a Point Source

• Since the intensity varies as 1/r2, this is an inverse square relationship

• The average power is the same through any spherical surface centered on the source

• To compare intensities at two locations, the inverse square relationship can be used

A small source emits sound waves with a power output of 80.0 W. (a) Find the intensity 3.00 m from the source. (b) At what distance would the intensity be one-

fourth as much as it is at r = 3.00 m? (c) Find the distance at which the sound level is

40.0 dB.(Ans: (a) 7.07 W/m 2(b) 6.00 m (c)1.00 x 10 – 8 W/m2 , 2.52 x 104 m)

Example 9:

Solution: