Standing Waves Flashcards
Wave Function for Standing Waves
y == 2A0 sin(kx) cos(ωt)
Standing Waves
Description
- every part of the string vibrates with SHM with the same frequency and phase
- amplitude is dependent on x
Standing Waves on a String
- natural patterns of oscillation of the string each with a different characteristic frequency
- the length of the string is always equal to an integer number of half wavelengths
- fundamental has the longest wavelength, λ/2=L
Harmonics and Overtones
- the fundamental is the lowest frequency mode
- frequency of each normal mode is an integer multiple of the fundamental frequency
- these integer multiples are called harmonics
- the fundamental is the first harmonic and so on
- these normal mode frequencies are also referred to as overtones, where the 2nd harmonic is the first over tone, the 3rd harmonic is the 2nd overtone so on
Standing Waves on a String
Description
when a stretched string is distorted so that the initial shape corresponds to a harmonic, only that particular normal mode is excited, so the string vibrates with the frequency of that harmonic
Standing Waves – Musical Instruments
Description
-when a string is struck/plucked on a piano/guitar, its initial shape is no that of a single normal mode. Several normal modes are excited resulting in vibration that includes several harmonic frequencies. Typically, the fundamental has the largest amplitude
Sound Waves
- longitudinal
- move through gases, liquids or solids
- speed depends on the properties of the medium
Classifying Sound Waves
Audible Waves - 20-20000 Hz
Infrasonic Waves - frequency below audible
Ultrasonic Waves - frequency higher than audible
Energy of a Sound Wave
Etot = 1/2 ρ(ωs0)² A vt
s0 = amplitude of sound wave A = surface area v = speed of wave
Sound Wave - Wave FUnction
S = S0 cos(ωt)
Maximum Velocity of a Sound Wave
Vmax = ωS0
Energy Density of a Sound Wave
1/2 ρω²S0²
Power
Definition (Sound Wave)
rate at which energy is transmitted to each layer
Power of a Sound Wave
P = 1/2 ρ(ωs0)² A v
Intensity of a Sound Wave
I = 1/2 ρ(ωs0)² v
Standing Waves in Air Columns
Open at Both Ends
- nodes at each end of the tube
- first normal mode, L = λ/2
- second normal mode, L = λ
- third normal mode, L = 3λ/2
- the integer numbers of half wavelengths of the normal modes are equal to the length
Standing Waves in Air Columns
Open at One End
- antinode at the closed end
- node at the open end
- first normal mode, L = λ/4
- second normal mode, L = 3λ/4
- third normal mode, L = 5λ/4
- for the normal modes L is equal to odd integer numbers of quarter wavelengths
Pressure Wave
Velocity Equation
v = √(B/p)
B = bulk modulus of compressibility p = density v = velocity
Bulk Modulus of Compressibility
Equation
B = γp
γ = Cp/Cv
Pressure Wave Equation
ΔP = BkS0 cos(kx-ωt)
Amplitude of Sound Wave
Pressure Equation
S0 = ΔP0 / ρωv
Sound Wave Intensity
Pressure Equation
I = 1/2 ρω²v {ΔP0/ρωv}²
Spherical Waves
- in a uniform medium, waves move outwards from the source at constant speed
- from a point source (a small object oscillating with SHM) are produced with spherical wavelengths
Intensity and Amplitude
I ∝ A²
-> I = cA²
A = √(I/c)
c = constant of proportionality
Decibels
-a lotharithmic scale
β = 10 log(10){I/I0)
β = measured in decibels (dB) I = Intensity of the sound I0 = threshold of hearing
