Pure Tones and the Sine Wave Physics of Music PHY103 Lecture 2 Image from www.math.ucdavis.edu/~angela/mathC.htmlTrig definition of a sine wave From math learning service U. Adelaide amplitude Period or wavelengthHarmonic motion From ecommons.uwinnipeg.ca/ archive/00000030/Velocity and Position Sine and Cosine From www2.scc-fl.com/lvosbury/AnimationsForTrigono... Sine and CosinePythagorean theorem Conservation of Energy When the spring is extended, the velocity is zero. When the spring is in the middle, the velocity is maximum. The position is the sine wave, the velocity is the cosine wave. Kinetic energy (square of velocity) + Potential energy (square of position) is total energy is conserved. a=c Cos(θ) b=c Sin(θ) θMaking a pure tone with MatlabSine Wave Period (units time or seconds) AmplitudeSine Wave Wavelength (units cm) Amplitude For a wave on water or on a string - spatial variation instead of temporal variation Position xAmplitude • Units depend on what is measured • Velocity, pressure, voltage? Angular frequency angular frequency radians per second frequency in Hz cycles per secondRelation between frequency and period • Suppose the period is P=0.2s • I need 5 cycles to add up to 1s • So the frequency is f=5Hz. • The number of periods/cycles that add up to 1 second is the frequency fP=1 f=1/PRelation between frequency and period P=1/f 1 second 12 cycles in 1 second The frequency is 12 Hz The period (time between each peak is 1/12 seconds or 0.083secondsHow does energy/power depend on the amplitude?How does energy/power depend on the amplitude? • Energy depends on the sum of the square of velocity and square of position (from equilibrium) • We expect that the energy or power (energy per second for a traveling wave) depends on the square of the amplitude. Power proportional to square of AmplitudeDecay – loss of energyShowing a sine wave on the oscilloscopeSignal or waveform generator Can adjust • Shape of wave (sine, triangle, square wave) • Voltage (amplitude of wave) • Frequency of wave Oscilloscope Adjust voltage of display (y–axis) Adjust time shown in display (x-axis) Adjust trigger Can also place display in x-y mode so can generator Lissajous figuresSine waves – one amplitude/ one frequency Sounds as a series of pressure or motion variations in air. Sounds as a sum of different signals each with a different frequency.Clarinet spectrum Clarinet spectrum with only the lowest harmonic remaining Time ! Frequency!Waveform view Full sound Only lowest harmonic Complex tone Pure toneTouching the string at a node after plucking harmonicDecomposition into sine waves • We can look at a sound in terms of its pressure variations as a function of time OR • We can look at a sound in terms of its frequency spectrum This is equivalent to saying each segment is equivalent to a sum of sine waves. “Fourier decomposition” Some of the character or “timbre” of different sounds comes from its spectrum: which harmonics are present, how strong they are, and where they are exactly (they can be shifted from integer ratios)Audition tutorial: Pulling up the spectral viewPlay Record Zoom in horizontal axis Loop Zoom in vertical axisRight click and hold on the axes will also allow you to adjust the rangeGetting a linear frequency spectrumHarmonics or OvertonesWavelengths and frequencies of Harmonics And velocity v on the stringRelation between frequency and wavelength quantity wavelength frequency meaning distance cycles per second units cm Hz symbol λ f wavelength of fundamental mode is inversely proportional to frequency longer wavelengths ! slower motionWavelength/Frequency cm x 1/s = cm/s frequency is related to wavelength by a speed -- The speed that disturbances travel down a stringTraveling wavesTraveling waves • Right traveling • Left traveling Law of cosinesSums of same amplitude traveling waves gives you standing wavesWhy the second mode has twice the frequency of the fundamental • Exciting the fundamental. Excite a pulse and then wait until it goes down the string and comes all the way back. • Exciting the second harmonic. When the first pulse gets to the end the string, you excite the next pulse. This means you excite pulses twice as often. You must drive at twice the frequency to excite the second modeAdding two traveling waves one moving left one moving right Standing wave!Traveling waves vs standing waves • Can think of standing waves as sums of left traveling and right traveling waves • The time to go from zero to max depends on the time for the wave to travel a distance of 1 wavelength! smaller wavelengths have faster oscillation periods (frequencies) • If the waves move faster on the string then the modes of oscillation (the standing waves) will be higher frequencyWave speed dimensional analysis • Only quantities we have available: – String density (mass per unit length) ρ – String length L – Tension on string T • Force = mass times acceleration F=ma (units g cm/s2) • Tension on a string is set by the force pulling on the string So T is units of g cm/s2 mgWave speed dimensional analysis continued • We want a velocity (cm/s). How do we combine the 3 physical quantities to get a velocity? – String density ρ (g/cm) – String length L (cm) – Tension T (g cm/s2) • T/ ρ has units cm2/s2 so a velocity is given by • To get a quantity in units of frequency we divide a velocity by a length • When we think about oscillating solids (copper pipes for example) the thickness is also important.Spring/String Spring String Heavier weight Slower frequency Heavier mass string slower fundamental mode frequency Stronger spring Higher frequency Tenser string Higher fundamental