##### ie-Physics
Waves and Energy Transfer

### summary and practice

##### © by William Dietsch 2006 #### Introduction:

Waves are a way in which energy is transferred from place to place without the transfer of matter.  The energy is carried from place to place in the form of a disturbance.

Waves, which move through matter, are called mechanical waves.  They require an elastic material medium through which to pass.  Examples of matter waves are water waves at the beach and sound waves moving through the air.  Waves, which do not require a material medium, are called electromagnetic waves.  Examples of electromagnetic waves are light and radio waves.

The wave-like behavior exhibited under certain conditions by particles like electrons are called matter waves.

#### Types of mechanical Waves:

Transverse waves are waves in which the movement of the vibrating particles is perpendicular to the direction of the wave.

Longitudinal waves are waves in which the movement of the vibrating particles is parallel to the direction of the wave.

Torsion waves are waves in which the movement of the vibrating particles is radial about the direction of the wave.

Combination waves are combinations of two or more of the above types.

#### Characteristics of Waves:

Wave speed (v) is the velocity of the wave as it moves through a medium.  Wave speed is also known as the propagation speed of the wave.  The speed of a wave is expressed in m/s.

Speed on a string (transverse): where F is the tension (N), m/L is the mass per unit length (Kg/m)

Speed through a metal rod (longitudinal): where Y is young’s modulus (N/m2) and ρ is the density (usually in units of Kg/m3).

Speed through a gas or liquid: Where β is the bulk modulus and ρ is the density.

Wavelength (λ) is the distance between corresponding points on adjacent wave pulses.  The wavelength of a wave is expressed in linear units, usually meters.

Frequency (ƒ) is the number of complete waves, which pass through a point in space per unit time.  The frequency of waves is measured in Hertz (Hz).  An older but more descriptive unit is cycles per second (cps).

The period (T) of a wave is the time required for a complete wave to pass through a point in space.  The period is the inverse of the frequency ( T = 1 / ƒ ).  The period of a wave is measured in seconds.

Amplitude (a) is the maximum displacement experienced by a particle as the result of a wave.  The amplitude is a measure of the energy carried by a wave.  The units for amplitude are specialized for the type of wave.  The amplitude of waves at the beach can be measured in meters.  The amplitude of sound waves is measured in a unit of pressure such as Pascals.  Amplitude is independent of the other wave characteristics.

The wave equationv = ƒ x λ ) expresses the relationship between wave speed (v), frequency ( ƒ ), and wavelength (λ).  The wave equation shows that frequency and wavelength are inversely proportional.  The wave equation is useful in many different applications and may be utilized in the analysis of matter waves as well as electromagnetic waves.

#### Behavior of Waves:

The reflection of waves is the turning back or changing of the direction of a wave when it encounters a boundary in the medium through which it is moving.  Waves follow the Law of Reflection, which states that the angle of incidence (incoming wave) is equal to the angle of reflection (outgoing wave), where the angles are measured from the normal (a line drawn perpendicular to the reflecting surface at the point of reflection.  If an incident wave encounters a boundary, which restricts the movement of the particles at the boundary, the reflected pulse is inverted.  If and incident wave encounters a boundary which does not restrict the movement of the particles, the reflected pulse is erect.

Refraction of waves is the bending of the wave's path as it passes obliquely (motion other than perpendicular to surface boundary) from one medium to another in which the waves have different propagation speeds, v.  The path of the wave bends toward the normal (perpendicular to surface) if the wave passes from fast speed to slow and vice versa.  Snell's law of refraction governs this: sin θi / sin θr = vi / vr where θ represents the respective angles of incidence i and after refraction r as measured from the normal.

Diffraction of waves occurs when waves spread out around corners or through openings in barriers.

Interference occurs when two or more waves pass simultaneously through the same medium.  The behavior of the medium is governed by the Principle of Superposition.  The principle of superposition states that: a) the individual wave characteristics (wavelength, amplitude, speed etc.) are unaffected by the other waves in the medium, and b) the displacement of the medium is the vector sum of the individual (component) waves.  When the resultant displacement of the medium is greater than any of the component waves, constructive interference occurs.  If the resultant displacement of the medium is less than any of the component waves, destructive interference occurs.  If the sum of the waves is equal to zero, complete destructive interference occurs.  Standing waves are an interference pattern caused by identical waves passing through a medium in opposite directions.  The parts of the pattern where there is no displacement (caused by complete destructive interference) are called nodes.  Nodes appear at 1/2 λ intervals.  For example air columns resonate in wind instruments and pipes so that their standard waves interfere producing their musical notes.

#### Sound Waves in matter

Sound is vibrations (of frequency ranging from 20 Hz to 20,000 Hz for a person with good hearing) in matter which are audible to human beings.  Sonic vibrations less than 20 Hz are called infrasonic and more than 20,000 Hz are called ultrasonic.  Sound waves consist of compressions (where the particles are closer together than normal) and rarefaction (where the particles are farther apart than normal).  The difference between compressions and rarefactions are usually measured in units of pressure such as Pascals.

The speed of sound in air can be calculated by using the equation: v = 331.4 + (0.606 T), where v is the speed of sound in air (m/s) and T is the air temperature in Celsius.

Pitch refers to the physiological response of a human being to the frequency of a sound wave.  Pitch "sounds" high if the frequency is rapid and "sounds" low if the frequency is slow.

Timbre refers to the quality of the sound produced by overtones (multiple frequency waves associated with the fundamental wave).  Timbre allows one to distinguish the different kinds of musical instruments since the relative size of the overtone waves depends on the instrument design.

Air column resonance occurs in tubes where standing waves are created.  The principal of air column resonance is how wind instruments such as trumpets, flutes and pipe organs produce sound.  The column may be open producing a wavelength, λ = 2 ( l + 0.8 d ) where l is the first resonant length, and d is the diameter of the tube, or closed producing λ = 4 ( l + 0.4 d ).  (If the diameter is not known, approximate using λ = 2 l for open pipes and λ = 4 l for closed pipes since most pipes are long and slender.)

Loudness is the physiological response of a human being to the intensity (power ≈ amplitude) of the sound wave.  A logarithmic scale called decibels, dB is used to measures loudness.  (This is a tenth of the unit named after Alexander Graham Bell, the founder of Bell Telephone Laboratory where the unit was created.)

The intensity level (in dB) = log10 ( I / Io )
Where I is the intensity (Watt/m2) and
Io is the weakest audible sound.

The Doppler effect occurs when the source and observer (or detector) are moving relative to each other.  In sound, the frequency (pitch) increases as the source approaches and decreases as the source recedes.  The detected frequency ƒd caused by the Dopper effect is given by

ƒd = ƒs + ƒs ( Δv / vs )

where ƒs is the original frequency produced by the source, vs is the speed of sound in air, and Δv is the relative speed caused by either motion of the observer, the source, or both.

#### Problems on Wave Equation

1. A ship rocks up and down with a frequency of 0.45 Hz.  Compute the period of the waves.

2. A wave has a length of 12 m and a period of 2.0 s.  Compute the velocity of the wave.

3. A water wave travels 4.5 m in 1.8 s.  The period of the wave is 1.2 s.  Compute the speed and wavelength of the wave.

4. The speed of light is 3.0 x 108 m/s.  What is the wavelength of a light wave with a frequency of 5.0 x 1014 Hz?

5. AM radio is broadcast at frequencies between 550 kHz and 1600 kHz.  Radio waves travel at the speed of light (3.0 x 108 m/s).  Compute the range of wavelengths of AM radio waves.

6. FM radio is broadcast at frequencies between 88 mHz and 108 mHz.  Radio waves travel at the speed of light (3.0 x 108 m/s). Compute the range of wavelengths of FM radio waves.

7. A sonar signal has a frequency of 1.00 x 106 Hz. and a wavelength of 1.50 mm in water.  What is the speed and period of these waves? How long will it take for the sonar waves to return from a contact, which is 1500 m away? (Round trip time).

8. The speed of sound in water is 1498 m/s. A sonar signal is sent out from a ship and returns 1.8  s later.  How far is the bottom of the ocean below the ship?

9. The speed of the transverse waves produced by an earthquake (the P waves) is 8.9 km/s and the speed of the longitudinal waves (the S waves) is 5.1 km/s.  If the P waves arrive at a seismograph 73 s before the S waves, how far away is the earthquake?

#### Superposition and Interference of Waves

1. Using 1/10-inch graph paper for convenience, sketch portions of the following wave pairs (e.g., A & B) adjacent to each other.  Use different colors for each two waves.  You may draw stair-step-like waves (_⌈⌉_⌈⌉_) or more realistic sine curves.
 WAVE A WAVE B λ = 4 inches A = 2 inches λ = 3 inches A = 2 inches WAVE C WAVE D λ = 6 inches A = 1 inch λ = 4 inches A = 1 inch WAVE E WAVE F λ = 4 inches A = 1.5 inches λ = 2 inches A = 2 inches WAVE G WAVE H λ = 2 inches A = 2.5 inches λ = 5 inches A = 1 inch WAVE I WAVE J λ = 1.5 inches A = 2 inches λ = 1 inch A = 1 inch
1. Add  A  +  B,      C  +  D,      E  +  F,      G  +  H,    and   I  +  J  by combining the amplitudes of the adjacent waves constructed in the previous problem. Considering these waves as a snapshot in time of waves passing on the same strand or surface while moving in opposite directions.  It might be worth moving each a fraction of an inch in the directions of their motions, then again constructing the sums, producing the equivalent of movie frames of the interference due to the two passing waves.

#### Problems on Sound Waves

1. Sound with a frequency of 445 Hz has a speed in water of 1435 m/s.  Compute the wavelength of the sound waves in water.

2. Compute the speed of sound in air at 41°C.

3. Find the frequency of a sound wave at 26°C with a wavelength of 0.681 m.

4. Humans can hear sound between 20 Hz and 20,000 Hz.  Compute the range of wavelength audible by humans at 10°C.

5. What is the frequency of sound at 15°C having a wavelength equal to the diameter of a 38 cm speaker known as a "woofer"?

6. What is the frequency of sound at 15°C having a wavelength equal to the diameter of a 7.6 cm speaker known as a "tweeter"?

7. A closed pipe has a first resonant length of 38 cm.  If the temperature of the air is 23°C, compute the frequency produced. Ignore the diameter correction in this problem.

8. A closed pipe resonates at a frequency of 366 Hz.  What is the first resonant length of the pipe when the speed of sound in air is 343 m/s?  Ignore the diameter correction in this problem.

9. An open pipe is to produce a frequency of 440 Hz when the speed of sound is 343 m/s.  Compute the first resonant length of the pipe. Ignore the diameter correction in this problem.

10. A child's whistle is 15 cm long, with a diameter of 1.25 cm and represents a closed pipe.  What frequency will it produce at 25°?

11. An open organ pipe is 1.23 m long and has a diameter of 10.0 cm.  What frequency will it produce at a temperature of 15.0°C?

#### Doppler Effect

1. A locomotive approaches a crossing at 95 Km/h.  Its horn has a frequency of 288 Hz and the air temperature is 15°C.  What is the frequency of the sound perceived by a crossing guard?

2. What is the frequency of the sound observed by a stationary person when a train moving away at 125 Km/h sounds its 320 Hz horn?  The temperature of the air is 25°C.

3. A bicycle rider approaches a factory whose whistle is blowing.  The temperature of the air is 23°C and the frequency of the whistle is 555 Hz.  How fast is the rider approaching the factory if he observes a frequency of 579.3 Hz?

4. If the above rider rides away from the factory at a speed of 27 m/s, what frequency does he observe?

5. A fire engine driving at 45 m/s sounds its 1,600 Hz siren as it approaches a car.  The car approaches the fire engine with a speed of 42 m/s.  The air temperature is 26°C.  What frequency will the driver of the car will observe?

6. The frequency of a siren on a fire station is 3500 Hz.  If you are driving away from the fire station at a speed of 40.0 Km/h and the air temperature is 19°C, what frequency will you hear?

7. A car approaching you at 72 Km/h sounds its 550 Hz horn.  What frequency will you hear if the temperature is 25°C.? created and © 2006 by William Dietsch
posted & edited 5 February 2008 by D Trapp