PHYSICS 149: Lecture 23 • Chapter 11: Waves – – – –

Lecture 23

11.5 Mathematical Description of a Wave 11.6 Graphing Waves 11.7 Principle of Superposition 11.8 Reflection and Refraction

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Midterm Exam 2 • • • • • •

Wednesday, November 17, 6:30 PM – 7:30 PM Place: PHY 114 Chapters 5 - 8 The exam is closed book. The exam is a multiple-choice test. There will be ~15 multiple-choice problems. – Each problem is worth 10 points. • Note that total possible score for the course is 1,000 points (see the course syllabus)

• The difficulty level is about the same as the level of textbook problems. • You may make a single crib sheet – you may write on both sides of an 8.5” × 11.0” sheet Lecture 20

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Simple Harmonic Motion • Occurs when having linear restoring force F= -kx – x(t) = [A] cos(ωt) – v(t) = -[Aω] sin(ωt) – a(t) = -[Aω2] cos(ωt)

• Springs – F = -kx – U = ½ k x2 – ω = sqrt(k/m)

• Pendulum (small oscillations) – ω = sqrt(L/g) Lecture 23

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Types of Waves • Transverse: The medium oscillates perpendicular to the direction the wave is moving. – Water (more or less) – Slinky

energy transport

• Longitudinal: The medium oscillates in the same direction as the wave is moving. – Sound – Slinky energy transport Lecture 23

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Waves on a String

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Speed of Transverse Waves • The speed of a transverse wave on a string is where No!

F: tension m: mass of string L: length of string μ: mass density

• The speed of a transverse wave depends on mechanical properties of the wave medium. More restoring force makes faster waves; more inertia makes slower waves. Lecture 23

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Periodic Waves • A periodic wave is a wave that repeats the same pattern over and over. amplitude: A

• Period T: the time for a single point to repeat itself (that is, the time for a single point to move from crest Æ equilibrium Æ trough Æ equilibrium Æ crest). Or, the time for a pulse to move from a crest to the next crest. • Frequency f: the number of cycles (for a single point) per unit time • Wavelength λ: the distance from a crest to the next crest • Amplitude A: the maximum displacement of a single point from its equilibrium position Lecture 23

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Example: Speed on a String • A 2 m long string has a mass of m = 6.40 g. What is the speed of transverse waves on this string when its tension is 90.0 N? Linear mass density: μ = m / L = (6.40 × 10-3 kg) / (2 m) = 3.20 × 10-3 kg/m Speed of transverse waves: v = sqrt(F/μ) = sqrt[ (90.0 N) / (3.20 × 10-3 kg/m) ] = 168 m/s Lecture 23

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Harmonic Waves • Harmonic waves: a special kind of periodic wave in which the disturbance is sinusoidal (either a sine or cosine function). • In a harmonic transverse wave on a string, every point on the string moves (in the direction perpendicular to the direction of propagation of the wave) in simple harmonic motion with the same amplitude and frequency, although different points reach their maximum displacement at different times.

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Harmonic Waves y(x,t) = A sin(ωt – kx) A = amplitude ω = angular frequency k = wave number = 2π/λ

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Amplitude and Wavelength y(x,t) = A cos(ωt – kx) Wavelength: The distance λ between identical points on the wave. Amplitude: The maximum displacement A of a point on the wave. Angular Frequency ω: ω = 2 π f Wave Number k: k = 2 π / λ Recall: f = v / λ

y

Wavelength

λ

Amplitude A A Lecture 23

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x

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Period and Velocity l

l

Period: The time T for a point on the wave to undergo one complete oscillation.

Speed: The wave moves one wavelength λ in one period T so its speed is v = λ / T.

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Period and Frequency •

Period T – The time for a single point to repeat itself, or the time for a pulse to move from a crest to the next crest – Units: s, min, and so on

•

Frequency f – The number of cycles (for a single point) per unit time – Unit: Hz (note that 1 Hz = 1 cycle/s)

•

Wavelength λ

– The distance from a crest to the next crest – Units: m, cm, and so on

•

Amplitude A – The maximum displacement (distance) of a single point from its equilibrium position – Units: m, cm, and so on

•

v=fλ

Relationships

– The intensity of a wave is proportional to the square of its amplitude (I ∝ A2). Lecture 23

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Example: Periodic Waves • What is the speed of a wave whose frequency and wavelength are 500.0 Hz and 0.500 m, respectively? And, what is the period of the wave? f = 500.0 Hz and λ = 0.500 m v = f⋅λ = (500.0 Hz)×(0.500 m) = 250 m/s T = 1/f = 1 / (500 Hz) = 2×10-3 s Lecture 23

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Harmonic Waves Exercise y(x,t) = A cos(ωt – kx) Label axis and tic marks if the graph shows a snapshot of the wave y(x,t) = 2 cos(4t – 2x) at x=0. Recall: T = 2 π /ω What is the period of this wave? T = 2 π / ω = 2 π/ 4 = 1.58 s +2

What is the velocity of this wave? π/4

-2

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π/2

3π/4

t v = ω/k= 4/2 m/s=2m/s

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Harmonic Waves • Direction of Travel:

– Wave travels in the +x-direction, if ωt and kx terms have the “different” signs. – Wave travels in the –x-direction, if ωt and kx terms have the “same” sign (that is, both positive or both negative).

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Direction

A wave y = A cos(ωt - kx) travels in +x direction A wave y = A cos(ωt + kx) travels in -x direction Δt

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ILQ •

What is the equation of a harmonic wave traveling to the left (–x direction) with amplitude 5.0 mm, angular frequency 65 rad/s, and wave number 4 rad/m ?

a) b) c) d)

y(x,t) = (5.0 mm)⋅sin[(65 rad/s)t + (4 rad/m)x] y(x,t) = (65 rad/s)⋅sin[(5.0 mm)t + (4 rad/m)x] y(x,t) = (5.0 mm)⋅sin[(65 rad/s)t – (4 rad/m)x] y(x,t) =(4 rad/m)⋅sin[(65 rad/s)t +(0.0050 m)x]

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Graphing Waves • If t is held constant, the graph shows an instantaneous picture of what the wave looks like at that particular instant (that is, a snapshot). • If x is held constant, the graph shows the motion of the (single) point x as a function of time (t).

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Wave Properties • The speed of a wave is a constant that depends only on the medium, not on amplitude, wavelength or period (similar to SHM) λ and T are related ! λ = v T or λ = 2π v / ω or λ = v / f

•

v=λ/T (since T = 2π / ω ) (since T = 1/ f )

Recall f = cycles/sec or revolutions/sec ω = 2πf

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ILQ •

A boat is moored in a fixed location, and waves make it move up and down. If the spacing between wave crests is 20 m and the speed of the waves is 5 m/s, how long does it take the boat to go from the top of a crest to the bottom of a trough?

a) b) c) d)

1 second 2 seconds 4 seconds 8 seconds

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Wave Description

λ – wavelength: distance between crests (meters) T – period: the time between crests passing fixed location (seconds) v – speed: the distance one crest moves in a second (m/s) f – frequency: the number of crests passing fixed location in one second (1/s or Hz)

ω – angular frequency: 2πf: (rad/s)

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ILQ 1 A rope of length L and mass M hangs from a ceiling. The rope is not an ideal rope. If the bottom of the rope is pulsed to cause a wave to travel up the rope, the speed of the wave as it travels up A) increases. B) stays the same. C) decreases.

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ILQ 2 Suppose a periodic wave moves through some medium. If the period of the wave is increased, what happens to the wavelength of the wave assuming the speed of the wave remains the same?

A) The wavelength increases B) The wavelength remains the same C) The wavelength decreases

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λ=vT

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Principle of Superposition • When two or more waves overlaps, the net disturbance at any point is the sum of the individual disturbances due to each wave. (ex. this can be seen by dropping two pebbles into a pond.)

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Interference and Superposition • When two waves overlap, the amplitudes add. • y(x,t) = y1(x,t) + y2(x,t) – Constructive: increases amplitude – Destructive: decreases amplitude

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ILQ •

Two waves travel towards each other with v = 10 cm/s. What will be the amplitude of the resultant wave 1 second from now?

a) b) c) d)

0.30 cm 0.50 cm 0.80 cm 1.30 cm

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Reflection • Reflection occurs at a boundary between different wave media. • Some energy may be transmitted into the new medium and the rest is reflected (that is, some or all of the energy will travel back into the first medium). • How much of the energy is reflected depends on how different the properties of the two media are (in particular, on the wave speed in the two media); the more different, the more reflection takes place. Lecture 23

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Reflection • When a wave travels from one boundary to another, reflection occurs. Some of the wave travels backwards from the boundary – Traveling from fast to slow inverted – Traveling slow to fast upright

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Reflection Act • A slinky is connected to a wall at one end. A pulse travels to the right, hits the wall and is reflected back to the left. The reflected wave is A) Inverted B) Upright – Fixed boundary reflected wave inverted – Free boundary reflected wave upright

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Reflected Wave: Inverted or Not? • The reflected wave will be inverted if it reflects from a medium with a higher mass density (that is, with a lower wave speed; recall ). • The reflected wave will not be inverted if it reflects from a medium with a lower mass density (that is, with a higher wave speed). Å This phenomenon can be explained by (1) The principle of superposition at the fixed point at the end, or (2) Newton’s third law for the forces between the string and the wall. Lecture 23

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