Mechanical Waves Chapter 16

Expectations After this chapter, students will: know what a mechanical wave is. distinguish between transverse and longitudinal waves. know how wavelength, period, and velocity are related for periodic waves. identify the frequency, wavelength, amplitude, and direction of travel of a wave from the wave equation.

Expectations After this chapter, students will: calculate the speed of a wave on a string. recognize sound as a longitudinal wave. relate power to sound intensity and intensity levels. apply Doppler effect calculations to situations involving moving sources of sound, or moving observers.

Waves: What Are They? A wave is a travelling condition or disturbance. Energy travels from one place to another by means of a wave. Transverse wave: disturbance is perpendicular to travel direction. Longitudinal wave: disturbance is parallel to travel direction.

Periodic Waves If the source of the disturbance produces it repeatedly, at equal time intervals, the resulting wave is called periodic. Like anything else periodic, these waves are characterized by an amplitude, a period, and a frequency.

Periodic Waves T

Periodic Waves

Periodic Waves Amplitude: maximum magnitude of disturbance Period: time required for one complete cycle Wavelength: distance required for one complete cycle Frequency: number of cycles per second of time

Periodic Waves Relationships:

The Wave Equation We can write an expression for the disturbance as a function of both position and time: This is called the wave equation. “+” if wave travels toward –x “-” if wave travels toward +x

The Wave Equation Follow the point on the wave where y = 0: we see that this wave is moving toward the right (+x).

The Wave Equation The wave equation for this rightward-moving wave: If we freeze time (constant t, “snapshot”): then we have y as a function of position (x), and ft is a constant phase angle whose value depends on the time at which we stopped the clock.

The Wave Equation The wave equation for this rightward-moving wave: Now, if we choose just one location (constant x): then we have y as a function of time (t), and fx is a constant phase angle whose value depends on the x we chose.

The Wave Equation If we see an equation that looks like: ... we can write down the amplitude, frequency, velocity, and wavelength of the wave it describes.

Speed of a Wave on a String A transverse wave on a string (or wire, rope, cable, etc.) depends on the tension in the string, as well as its diameter and the material from which it is made: tension force (string mass / string length)

Sound Sound is a longitudinal wave in which the disturbance is a change in the pressure in the air (or other medium).

Sound Like any wave, sound is characterized by a velocity and a wavelength.

Sound As with any wave, the disturbance travels, and energy travels, but the material (air) “sloshes back and forth” mostly in one place.

Sound: Speed The speed of a sound wave depends on the mechanical properties of the material through which it moves. Gas: Liquid: Solid:

Sound: Energetics The energy carried by a sound wave per second is its power: Power has SI units of J/s = W (watts)

Sound: Energetics We define the intensity of a sound wave as the power it carries perpendicularly through a surface, divided by the area of the surface: Intensity has SI units of W/m2. Intensity decreases from surface 1 to surface 2.

Sound: Energetics If the source of the sound wave radiates waves equally in all directions (spherically symmetric): sphere area

Sound: Energetics We can compare the intensities of two sound waves in terms of intensity levels: b is dimensionless, but is labeled with units of decibels (dB). I0 is a reference level: usually the “threshhold of hearing,” 1.0×10-12 W/m2 .

The Doppler Effect The Dopeler Effect is what happens when a stupid idea seems like a good idea because it comes at you really fast. But wait: we wanted to talk about the Doppler Effect, instead.

The Doppler Effect The Doppler Effect is the change in observed frequency of a sound wave (other sorts of waves, too) because of the movement of either the source, or the observer, or both, relative to the air through which the sound is traveling.

The Doppler Effect The Doppler Effect is the change in observed frequency of a sound wave (other sorts of waves, too) because of the movement of either the source, or the observer, or both, relative to the air through which the sound is traveling. The observer’s motion causes him to intercept more waves per second than he would if he were standing still.

The Doppler Effect Equations for a stationary source and moving observer: observer moves away from source observer moves toward source

The Doppler Effect General case (both source and observer move relative to the air): “+” if observer moves toward source; “-” if observer moves away from source “-” if source moves toward observer; “+” if source moves away from observer

Ch. 16 Takeaways Wavelength, frequency, period, velocity: Wave equation:

Ch. 16 Takeaways Transverse wave on string: Sound intensity: Spherically symmetric source: Sound intensity level:

Ch. 16 Takeaways Doppler effect: moving source, stationary observer: source moves away from observer source moves toward observer observed frequency source frequency speed of source speed of sound

Ch. 16 Takeaways Doppler effect: moving observer, stationary source: observer moves away from source observer moves toward source

Ch. 16 Takeaways Doppler effect, general (both source and observer move): “+” if observer moves toward source; “-” if observer moves away from source “-” if source moves toward observer; “+” if source moves away from observer