WAVES AND WAVE PHENOMENA Physics 12 Source: Giancoli Chapter 11

Objectives

Waves  What are some examples of waves in the “real” world?  Do these waves transmit energy or matter? How do you know?  How does the velocity of the particles that transmit the wave relate to the velocity of the wave itself?

Waves  The source of any wave is a pulse of energy.  If the pulse is consistent (as in a vibration) the wave becomes continuous.  What are some kinds of waves that you can think of?

Waves  Two common types of waves are mechanical waves and electromagnetic waves.  We will focus on mechanical waves.

Waves Important quantities that describe waves TermDefinition displacement, xthe movement of particles above and below the mean position amplitude, Athe difference between the maximum displacement and the mean position period, Tthe amount of time required to complete one cycle (between two identical positions) frequency, fthe number of complete cycles passing a given point in one second (Hz) wavelength, λthe distance covered in a complete wave cycle. wave velocity, vthe velocity at which wave crests move

Waves Wave motion Suppose the x- axis is time, t. How do we find the velocity, v, at which the wave moves?

Waves  A wave crest travels the distance of one wavelength, λ, in a time equal to one period, T.  Therefore, the wave velocity is v = λ/T or v = λf  What factors might affect the velocity of a wave?

Waves  The medium in which the wave travels is very important: think of sound through air and sound through water.  For a cord, the velocity varies accordingly: v = where F T is the tension in the cord, m/L is mass per unit length

Waves v =  What happens to v if the cord is tighter?  What happens if the cord is heavier? Why is this so?

Waves A wave whose wavelength is 0.30 m is traveling down a 300-m long wire whose total mass is 15 kg. If the wire is under a tension of 1000 N, what are the speed and frequency of the wave?

Waves v = 140 m/s f = 470 Hz

Types of waves  There are two main classifications of waves: 1. transverse 2. longitudinal  They differ in the relationship of the direction of particle motion and the direction of wave motion.

Types of waves Transverse wave What are some examples of a transverse wave?

Types of waves Longitudinal wave What is a common example of a longitudinal wave?

Mathematical description  The location of a point on a wave, y, depends on two variables—position and time (x and t), such that: y = A sin (ωt ± κx) where A is the amplitude, ω is the angular frequency (2πf) and κ is the angular wave number (2πk) k is the propagation constant

Superposition of waves  What happens when two waves meet?

Superposition of waves  When two waves meet, they are said to interfere with each other.  There are two types of interference: constructive and destructive.  When do you think each type of interference occurs?

Superposition of waves  When two waves have the same displacement (either both positive or both negative) when they meet, the resultant wave will be greater than each individual wave.  If the two waves are exactly in phase, maximum constructive interference will occur.  The amplitude of the resultant wave will be the sum of the two individual waves’ amplitudes.

Superposition of waves Constructive interference

Superposition of waves  When two waves have opposite displacements when they meet, destructive interference will occur.  If the waves are out of phase, then maximum destructive interference will occur.  The amplitude of the resultant wave will be the difference of the amplitudes of the individual waves.

Superposition of waves Destructive interference

Superposition of waves Two waves, one with an amplitude of 8 cm and the other with an amplitude of 3 cm, travel on a single string and overlap. What are the maximum and minimum amplitudes of the string while these waves overlap?

Superposition of waves 11 cm and 5 cm

Resonance  When you vibrate a cord at just the right frequency, you produce a standing wave.  It is called “standing” because the waves do not appear to be moving.  The points that are still are called nodes and the points with maximum amplitude are called antinodes.

Resonance Standing waves

Resonance  Standing waves can occur at numerous frequencies.  The frequencies at which standing waves occur are called natural frequencies or resonant frequencies.

Resonance Resonant standing waves (Harmonics) What is the relationship between L and λ in each standing wave?

Resonance L = where L is the length of the string and n is the number of the harmonic How do we determine the frequency f that these harmonics occur at?

Resonance If L =, then λ n = Furthermore, if f n = v n / λ n then f n = = = nf 1

Resonance A piano string is 1.10 m long and has a mass of 9.00 g. (a) How much tension must the string be under tot vibrate at a fundamental frequency of 131 Hz? (b) What are the frequencies of the first four harmonics?

Resonance (a) 679 N (b) 262 Hz, 393 Hz, 524 Hz