AP Physics 2- 14.2 Periodic Waves- Study Notes- New Syllabus
AP Physics 2- 14.2 Periodic Waves – Study Notes
AP Physics 2- 14.2 Periodic Waves – Study Notes – per latest Syllabus.
Key Concepts:
- Physical Properties of a Periodic Wave
Physical Properties of a Periodic Wave
A periodic wave is a continuous, repeating disturbance that transfers energy through a medium. Unlike a single wave pulse, a periodic wave consists of multiple oscillations generated by a continuous source.
Key Properties:
Amplitude (\(\mathrm{A}\)):
Maximum displacement from equilibrium. Determines wave energy (\(\mathrm{E \propto A^2}\)).
Wavelength (\(\mathrm{\lambda}\)):
Distance between two identical points on consecutive cycles (e.g., crest-to-crest).
Period (\(\mathrm{T}\)):
Time for one complete cycle. Related to frequency by \(\mathrm{f = \dfrac{1}{T}}\).
Frequency (\(\mathrm{f}\)):
Number of oscillations per second. Units: \(\mathrm{Hz}\).
Wave Speed (\(\mathrm{v}\)):
Related to frequency and wavelength: $ \mathrm{v = f \lambda} $
Phase:
Indicates the relative position within the oscillation cycle, measured in radians or degrees.
Mathematical Description of a Sinusoidal Wave:
Displacement as a function of time (at a fixed position \(\mathrm{x}\)):
$ \mathrm{y(t) = A \sin(\omega t + \phi)} $
where
\(\mathrm{\omega = 2\pi f = \dfrac{2\pi}{T}}\) is the angular frequency, and \(\mathrm{\phi}\) is the phase constant.
Displacement as a function of position (at a fixed time \(\mathrm{t}\)):
$ \mathrm{y(x) = A \sin(kx + \phi)} $ where \(\mathrm{k = \dfrac{2\pi}{\lambda}}\) is the wave number.
General wave equation (displacement as a function of position and time):
For a wave traveling in the +x direction:
$ \mathrm{y(x,t) = A \sin(kx – \omega t + \phi)} $
For a wave traveling in the –x direction:
$ \mathrm{y(x,t) = A \sin(kx + \omega t + \phi)} $
Example :
A periodic wave on a string is described by \(\mathrm{y(x,t) = 0.05 \sin(2\pi x – 4\pi t)}\), where \(\mathrm{y}\) and \(\mathrm{x}\) are in meters and \(\mathrm{t}\) in seconds. Find the amplitude, wavelength, frequency, and speed of the wave.
▶️ Answer/Explanation
Step 1: Amplitude: \(\mathrm{A = 0.05 \, m}\).
Step 2: Wave number: \(\mathrm{k = 2\pi / \lambda = 2\pi \Rightarrow \lambda = 1.0 \, m}\).
Step 3: Angular frequency: \(\mathrm{\omega = 4\pi \, rad/s \Rightarrow f = \dfrac{\omega}{2\pi} = 2 \, Hz}\).
Step 4: Wave speed: \(\mathrm{v = f \lambda = (2)(1.0) = 2.0 \, m/s}\).
Final Answer: \(\mathrm{A = 0.05 \, m, \; \lambda = 1.0 \, m, \; f = 2 \, Hz, \; v = 2.0 \, m/s}\).
Example :
A sinusoidal wave has a frequency of \(\mathrm{10 \, Hz}\), amplitude \(\mathrm{0.02 \, m}\), and wavelength \(\mathrm{0.50 \, m}\). Write the general wave equation if it travels in the +x direction and passes through equilibrium at \(\mathrm{t = 0}\), \(\mathrm{x = 0}\).
▶️ Answer/Explanation
Step 1: Angular frequency: \(\mathrm{\omega = 2\pi f = 2\pi (10) = 20\pi \, rad/s}\).
Step 2: Wave number: \(\mathrm{k = \dfrac{2\pi}{\lambda} = \dfrac{2\pi}{0.50} = 4\pi \, rad/m}\).
Step 3: Amplitude: \(\mathrm{A = 0.02 \, m}\).
Step 4: Since the wave passes equilibrium at \(\mathrm{x=0, t=0}\), phase \(\mathrm{\phi = 0}\).
Final Equation: $ \mathrm{y(x,t) = 0.02 \sin(4\pi x – 20\pi t)} $