Vibrations and Waves Summary Sheet Chapters 11 and 12 Simple Harmonic Motion Notation A = amplitude f = frequency Amplitude (A), max PE T = period m Equilibrium (x = o), max KE x = displacement m m m m v = speed Amplitude (A), max PE m m m a = acceleration t = time 2 2 1/2 1/2 1/2 x = A sin(2πft) v = vmax(1-x /A ) T = 2π(m/k) T = 2π(l/g) m = mass 1/2 or vmax = A (k/m) f = 1/T Note: period not F = force on spring k = spring constant x = A cos(2πft) 2 2 dependent on mass a = amaxcos(2πft) Etotal = ½ kx + ½ mv or amplitude l = pendulum length a = kA/m 2 xmax = A max F = kx g = 9.8m/s Anatomy and Types of Waves Wave Speed v = λf general equation v = wave speed wavelength crest wavelength compressions λ = wavelength E Using material f = frequency amplitude v = properties ñ E = elastic modulus (longitudinal) ρ = density FT = tension in string extensions trough F Waves on a l = length of string v = T m string m = mass of string Transverse Wave Longitudinal Wave (transverse) l Wave Properties Reflection – upon encountering a new medium, a pulse or wave may “bounce” back Ex ample: Example: water Angle of incidence θi Pulses on a Fixed end Free end waves hitting a equals angle of string barrier θr reflection Transmission –upon encountering a new medium, a portion of the wave continues into the new medium Example: Less dense to more dense: amplitude decreases, wave slows Pulses on string More dense to less dense: amplitude increases, wave speeds up Interference – occurs when multiple waves interact Principle of superposition – to find the resulting wave, the displacements of each wave are added Constructive – resulting amplitude is greater than either Destructive – resulting amplitude is less than pulse’s/wave’s amplitude either pulse’s/wave’s amplitude Refraction – wave changes direction due to a Diffraction – wave bends when it encounters change in the medium through which it travels a barrier Vibrations and Waves Summary Sheet Chapters 11 and 12 Standing Waves; Resonance Damped Harmonic Motion In class we saw that if you fix one end of a string or long spring and send a wave down from the other end the wave reflects and interferes with the wave being sent. At particular frequencies we observed a special event – the wave appeared to be standing rather than moving. Curve A represents overdamping – system is Fundamental or brought to equilibrium over a long period of time. 1st harmonic Curve B represents critical damping – system is brought to equilibrium over a shorter period of time. Curve C represents underdamping – system undergoes several oscillations before reaching st equilibrium. 1 overtone or 2nd harmonic Often we design for critical damping – situation that brings the system back to equilibrium in a short period of time without any oscillations. Example: the car shock absorber. nd 2 overtone or rd 3 harmonic Doppler Effect Observers are stationary; sound source is moving f f ! = new frequency f ( = v 1' s f = source frequency v vs = source speed f f ( = vo = observer speed vs 1+ v = wave speed v Sound source is stationary; observer is moving & v # f ( = $1+ o ! f % v " v & o # f ( = $1' ! f % v ".