Travelling and Standing Waves Many biological phenomena are cyclic. Heartbeats Circadian rhythms, Estrus cycles Many more e.g. s Such events are best described as waves. Therefore the study of waves is a major component of this biophysics course. Sound Waves: Study guide 2 (Acoustics, vibrating strings) Electromagnetic Waves: Study guide 3-4 (Light, X-rays, Infra-red) Quantum Mechanical Waves: Study guide 4-6 (motion of elementary particles/waves) 1
MATERIAL TO READ Web: http://www.physics.uoguelph.ca/biophysics/ www.physics.uoguelph.ca/tutorials/gof/ /tutorials/trig/trigonom.html /tutorials/shm/q.shm.html Text: chapter 1, Vibrations and waves Handbook: study guide 1 For 1 st quiz, you should know: 1. How to evaluate sin(x), cos(x), sin -1 (x), cos -1 (x) 2. Equations for travelling and standing waves 3. How to graph a wave given the equation 4. Relations between period (T), frequency (f or υ) and wavelength (λ) Type of wave Form Sinusoidal y = Asin(x) Square y = A for 0 < x < λ/2, = -A for λ/2 x λ Ramp y = Cx, for -λ/2 x λ/2 Periodic functions can be described as a sum of sinusoidal waves. Therefore we need trigonometry 2
Trigonometry Review Trig. Functions defined on a circle P Point P moves around r s a circle of radius r θ y O x s θ (arc length) q = s/r dimensionless unit: See Web tutorial radians to review dimensions Common unit is degrees : Full circle = 360 o For full circle, s = circumference = 2πr \ s/r = 2p = 360 o 1 radian = 360 o /(2p) = 57.3 o. 3
Definitions: sin(θ) y/r Opposite/hypotenuse SOH Cos(θ) x/r Adjacent/hypotenuse CAH tan(θ) y/x Opposite/adjacent TOA Be careful calculating these functions using calculator: DEG or RAD mode on calculator indicates units. No degree symbol Examples: sin(40.0 o ) = 0.642 o means angle is in radians cos(2.5) = -0.801 As P moves around circles sin(θ) and cos(θ) vary 2 1 π/2 = -90 o π/2 = 90 o π = 180 o π = 360 o sin(θ) 0-1 -2-6 -4-2 0 2 4 6 8 10 θ 4
2 1 π/2 = -90 o π/2 = 90 o π = 180 o π = 360 o cos(θ) 0-1 -2-6 -4-2 0 2 4 6 8 10 θ NOTES: 1. Positive angles are measured counterclockwise from positive x axis 2. Both sinθ and cosθ repeat once every time P rotates around circle (2π radians) i.e. for any integer n, sin(θ+2πn) = sin(θ) Periodicity cos(θ+2πn) = cos(θ) 3. Also: sin(-θ) = -sin(θ) ODD cos(-θ) = cos(θ) EVEN 5
Small Angle Approximation: For θ < π/20 radians 10 o : θ MUST BE IN RADIANS sinθ θ cosθ 1 tanθ = sinθ/cosθ θ Not used until Study Guide 4. Inverse Trig. Functions: Inverse Sine sinθ = f (given f) what is θ? Imagine there is an inverse function for sin. Let s say it is called sin -1. Let s apply this function to both sides of the equation. sin -1 (sin(θ)) = sin -1 (f) θ = sin -1 (f) LOOK! THERE IS SIN-1 ON YOUR CALCULATOR 6
sin-1 also called arcsin(f) similar functions for cos -1 (f) and tan -1 (f). Example: Calc. setting sin -1 (0.683) = 0.752 (R: radians) = 43.1 o (D: degrees) Due to the periodic nature of sin, cos and tan there are actually an infinite set of angles that would produce the same value of each function 2 sin(θ) 1 0-1 f NB -1 < f < 1-2 0 2 4 6 8 10 θ All 4 values of θ have same f; only 2 per revolution. 7
Therefore sin -1 (f) is defined on interval 0 f 2π. The two values are : θ AND (π - θ) = (180 o - θ). The reason there are two values per revolution can be understood with the aid of a diagram. y θ 2 y 1 y 2 θ 1 x y 1 = sin(θ 1 ) = sin(θ 2 ) = y 2 Example: If sinθ = 0.45, θ =? Calculator returns θ = sin -1 (0.45) = 26.7 o Another value is: 180 o 26.7 o = 153 o THIS ALSO WORKS IF q IS NEGATIVE Flip the above diagram around to see it. Example θ = sin -1 (-0.48) 8
Inverse Cosine: 2 θ 1 θ 2 1 cos(θ) 0-1 -2 0 2 4 6 θ Example: θ = cos -1 (0.45) = 63.3 o What is other value? y x 1 θ 2 θ 1 x x 2 x 1 = cos(θ 1 ) = cos(θ 2 ) = x 2 9
From the diagram we see that the other value for θ is, θ 2 = 360 o - θ 1 = 296.7 o = -θ 1 NB! If we flip the above diagram, we see that this also works for θ < 0. Therefore cos -1 (f) is also defined on interval 0 f 2π. Its 2 values are : θ AND (2π - θ) = (360 o - θ). Inverse tangent: 2 1 tan(θ) 0 θ 1 θ 2-1 -2-2 0 2 4 6 θ Example: θ = tan -1 (1.0) = 45.0 o What is other value? 10
y θ 2 θ 1 x x 1 /y 1 = tan(θ 1 ) = tan(θ 2 ) = x 2 /y 2 From the diagram we see that the other value for θ is, θ 2 = θ 1 180 o = -135.0 o = 225.0 o. If we flip the above diagram, we see that this also works for θ < 0. Therefore tan -1 (f) is also defined on interval 0 f 2π. Its 2 values are : θ AND (θ - π) = (θ-180 o ). See Study Guide for more examples to practice on 11
Oscillations and Travelling Waves e.g. hanging weight on spring y y = 0 Equilibrium Position y = y 0 Initial position Spring stretched to initial position, y 0, and released. The mass oscillates between y 0 and -y 0 in "Simple Harmonic Motion" or SHM. Equation of motion: 12
ΣF = ma -ky = ma y (Ideal Spring Approx.) -(k/m)y = (d 2 y/dt 2 ) Diff. Equation Let's try: y(t) = y 0 sin(ωt) dy/dt = ωy 0 cos(ωt) where and w = 2p/T = angular frequency T = period = time for one cycle. NB: y(t) = y 0 sin(2π(t/t)) repeats every T Argument MUST be in radians d 2 y/dt 2 = -ω 2 y 0 sin(ωt) d 2 y/dt 2 = -ω 2 y(t) y(t) = y 0 sin(ωt) is solution to diff. eq. if ω 2 = k/m. This is why SHM is sinusoidal. Other e.g.'s: sound, light, water surface ripples. Looks like: T y 0 time t -y 0 13
Other useful definitions: Frequency = f, or n = number of cycles or oscillations per unit time (usually seconds). e.g.: Let, T = 0.5 s (period) in one second 2 oscillations occur that is: ν = 2 cycles/second u = 1/T = w/2p Dimensions of ν: (time) -1 Unit: s -1 or more explicitly, "cycles per second" Hertz So we can write equivalent representations: y(t) = y 0 sin(ωt) y(t) = y 0 sin(2πt/t) y(t) = y 0 sin(2πνt) 14
Travelling Waves e.g.: ripples on surface of water. Displacement is time and position dependent. Let s take snapshots of y vs. x at three successive times: (earliest, middle, latest) peaks moving towards right x (position) each point moves ONLY vertically y as a function of x at any fixed time is sinusoidal l y 0 x -y 0 l wave repeats itself after distance, l = wavelength. 15
y as a function of t at any fixed position is also sinusoidal T y 0 x -y 0 T In one period T, wave moves forward 1 wavelength. speed of Wave = Distance/time = λ/t = λν \v = ln governs all wave motion (water ripples, sound, light ) for velocity must also provide direction along +ve or ve x. Equations for Travelling Wave (Function of t, x) y = y 0 sin(ωt ± kx) + or - according to direction where k = 2p/l = wave number y = y 0 sin(2πt/t ± 2πx/λ) y = y 0 sin(2πνt ± kx) y = y 0 sin(k(vt ± x)) since kv = (2π/λ)v = 2πν = ω 16
If we look a t a particular position, equation reduces: e.g. x = 0 y = y 0 sin(2πt/t). y T/4 T/2 T 3T/4 2T t Alternatively can take snapshots at particular time, equation reduces again: y = y 0 sin(-2πx/λ) (we have assumed +ve x velocity for wave) y l/4 l/2 l 3l/4 2l x Wave direction? Crest of wave y = y 0 y = y 0 sin(2πt/t ± 2πx/λ) 1 = sin(2πt/t ± 2πx/λ) nπ + π/2 = (2πt/T ± 2πx/λ) (for n = 0, 1,2, ) Let s use n = 0 case and solve for x (position of crest) vs. t. π/2-2πt/t = ± 2πx/λ 17
± x = λ/4 - λt/t ± x = λ/4 - vt since λ/t = λν = v This separates into two equations: Upper sign x = λ/4 vt -ve velocity Lower sign and - x = λ/4 vt x = vt - λ/4 +ve velocity Therefore, in the equation: y = y 0 sin(2πt/t ± 2πx/λ) The upper sign corresponds to wave crest velocities in the negative x direction and the lower sign to those in the positive x direction. 18
Sample Problem example 1-5 from the text A wave moves along a string in the +x direction with a speed of 8.0 m/s, a frequency of 4.0 Hz and amplitude of 0.050 m. What are the (a) wavelength? (b) wave number? (c) period? (d) Angular frequency? (e) Determine an equation for this wave (and sketch it for t = 0 and t= 1/16 s) (f) What is value of y for the point x = ¾ m a t time t = 1/8 s? Solution (a) v = νλ v/ν = λ = 8.0/4.0 = 2.0 m (b) k = 2π/λ = 2π/2 = π m -1 (c) T = 1/ν = 1/4.0 = 0.25 s (d) ω = 2πν = 8π s -1 (e) y = y 0 sin(ωt ± kx) = 0.05 sin(8πt - πx) -ve sign given by fact that wave travels in +ve x direction (see previous page. For t = 0, y = 0.05 sin(- πx) = - 0.05 sin(πx) : 19
y (m) -0.05 1.0 2.0 l x (m) for t =1/16 s, y = 0.05 sin(π/2 - πx) = - 0.05 sin(π(x ½)) y = -0.05 sin(πx ), same as above except x = x - ½. y (m) 1.0 2.0 x (m) -0.05 redraw horizontal axis using x + ½ = x. y (m) 0.5 1.5 2.5 x (m) (f) y =0.05 sin(8πt - πx), for t =1/8 s and x = ¾ m. y =0.05 sin(π - 3π/4) = 0.05 sin(π/4) = 0.035 m 20
Sample Problem Self-test III, #1 c Sketch a graph for y = 4 sin(3πt - 6πx) at t = 0.900 s. y = 4 sin(3π(0.900) - 6πx) = 4 sin(2.7π - 6πx) = = -4 sin(6πx - 2.7π) = -4 sin(6π(x (2.7/6)π)) y = -4 sin(6π(x 0.45π)) = -4 sin(6πx ) x = x 0.45 k = 6π = 2π/λ λ = 2π/6π = 1/3 m y (m) 4 1/6 1/3-4 Redraw using x = x + 0.45 y (m) 1/6+0.45 1/3+0.45 = 0.62 m = 0.78 m= x (m) x (m) x = 0.45 m 21
Sample Problem Self-test III, #1 d Sketch y = 4 sin(3πt - 6πx) at x = 0.5 m (modified from x = 0.4 m). Since x is fixed plot y vs. t y = 4 sin(3πt - 6π(0.5)) = 4 sin(3πt - 3π) y = 4 sin(3π(t-1)) = 4 sin(3πt ) t = t 1.0 ω = 3π = 2π/T T = 2π/3π = 2/3 s y (m) 4 1/3 2/3-4 t (s) Redraw using t = t + 1.0 4/3 5/3 t (s) x = 1.0 m 22
Standing Waves Wave crests do not move e.g.: Vibrating string, sound in organ pipes Consider first: Reflection of travelling wave from surface x Incident wave Wave inverts upon reflection Reflected wave 23
A standing wave is the sum of two travelling waves with opposing velocities. Incident wave (towards x) y 1 = y 0 sin(ωt + kx) Reflected wave (towards +x) y 2 = - y 0 sin(ωt - kx) Inversion of wave Superposition principle: Resultant wave is the sum of the incident and reflected wave. y tot = y 1 + y 2 = y 0 [sin(ωt + kx) sin(ωt - kx)] Use trig. identity: sin(α ± β) = sinαcosβ ± cosαsinβ y tot = y = y 0 [ sin(ωt)cos(kx) + cos(ωt)sin(kx) - sin(ωt)cos(kx) + cos(ωt)sin(kx)] 0 y = 2y 0 [cos(ωt)sin(kx)] Equation for a STANDING WAVE Product of a time dependent term, (cos(ωt)), and a position dependent term, sin(kx). 24
y can be expressed as: y = a(t)sin(kx), where a(t) = 2y 0 cos(ωt) 2y 0 y loop = segment between nodes x -2y 0 node antinode Sketch of standing wave, y vs. x at different times a(t) > 0 a(t) < 0 At all times, node positions (y = 0) are at same x At all times, antinode positions are at same x 25
Sample Problem An incident wave y 1 = 3sin(2t - 3πx) and a reflected wave y 2 = -3sin(2t + 3πx) produce a standing wave (x, y, t in SI units). a) Write the equation of the standing wave. Using the summation identity as shown previously we can write the following: General Prescription: Incident wave: y 1 = y 0 sin(ωt - kx) Reflected wave: y 2 = - y 0 sin(ωt + kx) Standing wave: y = -2y 0 cosωtsinkx In this case we obtain: y = -6cos2t sin3πx b) What are the (i) amplitude standing wave s (ii) period (iii) and wavelength? (i) amplitude = 2y 0 = 6 m. (ii) ω = 2 s -1 = 2π/T T = π s (iii) k = 3π m -1 = 2π/λ λ = 2/3 m 26
c) Sketch the standing wave at times t = 0 s, t = T/4 and t = T/2. At t = 0 y = -6cos2t sin3πx y = -6cos2(0) sin3πx = -6sin3πx y (m) 6 1/3 2/3-6 x (m) = 0 At t = T/4 = p/4 s y = -6cos2t sin3πx y = -6cos(2π/4) sin3πx y = 0 for all x y (m) 6 1/3 2/3 x (m) -6 =-1 At t = T/2 = p/2 s y = -6cos2t sin3πx y = -6cos(2π/2) sin3πx y = 6 sin3πx (opp. of t = 0 graph) 27
Physics diagram aids 28