Waves and Energy Transfer

**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**.

**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.

**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/m^{2}) and ρ is the density (usually in units of Kg/m^{3}).

**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 equation** ( **v = 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.

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 θ_{i} / sin θ_{r} = v_{i} / v_{r} 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 **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

**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

The intensity level (in *dB*) =** log _{10} ( I / I_{o} )**

Where

** ƒ _{d} = ƒ_{s} + ƒ_{s} ( Δv / v_{s} )**

- A ship rocks up and down with a frequency of 0.45 Hz. Compute the period of the waves.
- A wave has a length of 12 m and a period of 2.0 s. Compute the velocity of the wave.
- 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.
- The speed of light is 3.0 x 10
^{8}m/s. What is the wavelength of a light wave with a frequency of 5.0 x 10^{14}Hz? - AM radio is broadcast at frequencies between 550 kHz and 1600 kHz. Radio waves travel at the speed of light (3.0 x 10
^{8}m/s). Compute the range of wavelengths of AM radio waves. - FM radio is broadcast at frequencies between 88 mHz and 108 mHz. Radio waves travel at the speed of light (3.0 x 10
^{8}m/s). Compute the range of wavelengths of FM radio waves. - A sonar signal has a frequency of 1.00 x 10
^{6 }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). - 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?
- 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?

- 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 |

- 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.

- 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.
- Compute the speed of sound in air at 41°C.
- Find the frequency of a sound wave at 26°C with a wavelength of 0.681 m.
- Humans can hear sound between 20 Hz and 20,000 Hz. Compute the range of wavelength audible by humans at 10°C.
- 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"?
- 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"?
- 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.
- 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.
- 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.
- 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°?
- 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?

- 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?
- 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.
- 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?
- If the above rider rides away from the factory at a speed of 27 m/s, what frequency does he observe?
- 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?
- 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?
- A car approaching you at 72 Km/h sounds its 550 Hz horn. What frequency will you hear if the temperature is 25°C.?

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