Physics 1B03summer-Lecture 9 Sine Waves f(x) = y(x,0) = A sin(kx) For sinusoidal waves, the shape is a sine function, eg., Then, at any time: y (x,t) = f(x – vt) = A sin[k(x – vt)] A -A y x ( A and k are constants)
Physics 1B03summer-Lecture 9 Sine wave: A -A y x v (“lambda”) is the wavelength (length of one complete wave); and so (kx) must increase by 2π radians (one complete cycle) when x increases by. So k = 2 , or k = 2π / λ y (x,t) = A sin[kx – t] since kv=ω
Physics 1B03summer-Lecture 9 Rewrite: y = A sin [kx – kvt]=A sin [kx – t] ω = 2πf y = A sin [ constant – t] “angular frequency” radians/sec frequency: cycles/sec (=hertz) The displacement of a particle at location x is a sinusoidal function of time – i.e., simple harmonic motion: The “angular frequency” of the particle motion is =kv ; the initial phase is kx (different for different particles). Review: SHM is described by functions of the form y(t) = A cos( t+ ) = A sin( /2 – – t), etc., with
Physics 1B03summer-Lecture 9 Example A -A y x Shown is a picture of a wave, y=A sin(kx t), at time t=0. a b c d e i) Which particle moves according to y=A cos( t) ? A B C D E ii) Which particle moves according to y=A sin( t) ? A B C D E iii ) Sketch a graph of y(t) for particle e.
Physics 1B03summer-Lecture 9 y(x,t) = A sin (kx ± t – ) angular wave number k = 2π / λ ( radians/metre ) angular frequency ω = 2πf (radians/second) amplitude “phase” The most general form of sine wave is y = Asin(kx ± ωt – ) The wave speed is v = 1 wavelength / 1 period, so v = fλ = ω / k phase constant
Physics 1B03summer-Lecture 9 Transverse waves on a string: (proof from Newton’s second law – Pg.625) Electromagnetic wave (light, radio, etc.): v = c 2.998 10 8 m/s (in vacuum) v = c/n (in a material with refractive index n ) (proof from Maxwell’s Equations for E-M fields ) The wave velocity is determined by the properties of the medium: Wave Velocity
Physics 1B03summer-Lecture 9 A) it will decrease by 4 B) it will decrease by 2 C) it will decrease by √2 D) it will stay the same E) it will increase by √2 You double the diameter of a string. How will the speed of the waves on the string be affected? Quiz
Physics 1B03summer-Lecture 9 Exercise (Wave Equation) What are and k for a 99.7 MHz FM radio wave?
Physics 1B03summer-Lecture 9 Particle Velocities Particle displacement, y (x,t) Particle velocity, v y = dy/dt ( x held constant) (Note that v y is not the wave speed v – different directions ! ) Acceleration, This is for the particles (move in y), not wave (moves in x) !
Physics 1B03summer-Lecture 9 “Standard” sine wave: maximum displacement, y max = A maximum velocity, v max = A maximum acceleration, a max = 2 A Same as before for SHM !
Physics 1B03summer-Lecture 9 Example string: 1 gram/m; 2.5 N tension Oscillator: 50 Hz, amplitude 5 mm y x Find: y (x, t) v y (x, t) and maximum speed a y (x, t) and maximum acceleration v wave
Physics 1B03summer-Lecture 9 1)Identical waves in opposite directions: “standing waves” 2)2 waves at slightly different frequencies: “beats” 3)2 identical waves, but not in phase: “interference” Superposition of Waves
Physics 1B03summer-Lecture 9 Practical Setup: Fix the ends, use reflections. node L (“fundamental mode”) We can think of travelling waves reflecting back and forth from the boundaries, and creating a standing wave. The resulting standing wave must have a node at each fixed end. Only certain wavelengths can meet this condition, so only certain particular frequencies of standing wave will be possible. example:
Physics 1B03summer-Lecture 9 λ2λ2 Second Harmonic Third Harmonic.... λ3λ3
Physics 1B03summer-Lecture 9 In this case (a one-dimensional wave, on a string with both ends fixed) the possible standing-wave frequencies are multiples of the fundamental: f 1, 2f 1, 3f 1, etc. This pattern of frequencies depends on the shape of the medium, and the nature of the boundary (fixed end or free end, etc.).
Physics 1B03summer-Lecture 9 Sine Waves In Opposite Directions: y 1 = A o sin(kx – ωt) Total displacement, y(x,t) = y 1 + y 2 y 2 = A o sin(kx + ωt) Trigonometry : Then:
Physics 1B03summer-Lecture 9 Example 8mm y 1.2 m f = 150 Hz x a)Write out y(x,t) for the standing wave. b)Write out y 1 (x,t) and y 2 (x,t) for two travelling waves which would produce this standing wave. wave at t=0
Physics 1B03summer-Lecture 9 Example When the mass m is doubled, what happens to a) the wavelength, and b) the frequency of the fundamental standing-wave mode? What if a thicker (thus heavier) string were used? m
Physics 1B03summer-Lecture 9 Example a) m = 150g, f 1 = 30 Hz. Find μ (mass per unit length) b) Find m needed to give f 2 = 30 Hz c) m = 150g. Find f 1 for a string twice as thick, made of the same material. m 120 cm
Standing sound waves Sound in fluids is a wave composed of longitudinal vibrations of molecules. The speed of sound in a gas depends on the temperature. For air at room temperature, the speed of sound is about 340 m/s. At a solid boundary, the vibration amplitude must be zero (a standing wave node). node antinode Physical picture of particle motions (sound wave in a closed tube) graphical picture
Physics 1B03summer-Lecture 9 Standing sound waves in tubes – Boundary Conditions -there is a node at a closed end -less obviously, there is an antinode at an open end (this is only approximately true) node antinode graphical picture
Physics 1B03summer-Lecture 9 L Air Columns: column with one closed end, one open end
Physics 1B03summer-Lecture 9 Exercise: Sketch the first three standing-wave patterns for a pipe of length L, and find the wavelengths and frequencies if: a)both ends are closed b)both ends are open