6.3: Periodic Waves
First off, what even is a periodic wave? Don’t panic, it’s not too complicated!
A periodic wave is a wave that has a repeating continuous pattern; it’s essentially a wave that repeats for several cycles. It can be characterized by its wavelength, period, frequency, and amplitude. All the waves below can be considered periodic waves.
Taken from Wikimedia Commons
The Wave Equation:
In Physics 2, we mainly focus on simple periodic waves that can be described with equations involving a sine or cosine function, as well as the wavelength, amplitude, and frequency of the wave.
As a function of time, the equation would look like:
f(t)=A cos(ω*t) ⏩ ω=2 π/Tω=2 π/T
F(t)=Acos(2π/T*t) ⏩ T=1/fT=1/f
As a function of position, the equation would look like:
F(x)=A cos (2π/λ*x)
ω= Angular Frequency
Let's practice together a little.
Take a look at this graph, where the x-axis is the horizontal position and the y-axis is the vertical height of the wave.
What's the general equation for periodic waves that are graphed as a function of position?
F(x)=A cos (2π/λ*x)
What's the amplitude for this wave?
What's highest/lowest point on the graph? 4
What's the wavelength? (Measure the distance from crest to crest, or trough to trough)
And that's it! The equation for this wave would look like:
F(x)=A cos (2π/λ*x) ➡️ F(x)=4cos(2π/3*x)
1.The frequency of the tuning fork is (approximately)
B) 0.020 s
C) 2.55 Hz
D) 50 Hz
E) 256 Hz
2. What is the amplitude of the wave?
(A) 4 cm
(B) 5 cm
(C) 8 cm
(D) 10 cm
(E) 16 cm
3. What is the speed of the wave?
(B) 25 cm/s
(C) 50 cm/s
(D) 100 cm/s
(E) 200 cm/s
4. What is the length of the longest wavelength shown?
A) 0.5 m
B) 0.75 m
C) 1 m
D) 2 m
E) 4 m
5. Which organ pipe(s) shows a standing wave which has twice the frequency of one of the other waves shown?
E) Cy , Cz, Ox, Oy
E: f = cycles / seconds
A: By inspection
C: By inspection, the λ is 10 cm. f = 1 / T = 5, Then use v = f λ.
E: Cx is only ¼ of a wavelength. To make a full wavelength you would need 4x the current length
D: Wavelengths of each are (dist/cycle) … 4L, 4/3 L, 4/5 L, L, 2/3 L …
Frequencies are f = v/ λ.
v/4L, 3v/4L, 5v/4L, v/L, 3v/2L … Oy is 2x Cy