The document provides an overview of standing waves, explaining their formation through the interference of two waves traveling in opposite directions with equal amplitude, wavelength, and frequency. Key concepts include nodes and anti-nodes, the mathematical representation of standing waves, and the concept of fundamental frequency in strings with fixed ends. A practical example involving the calculation of a guitar string's mass using the third harmonic frequency demonstrates the application of these principles.

1. STANDING WAVES
Physics 101 - Ch. 14-12

2. PREVIOUSLY…
• We took a look into traveling waves
• Traveling waves are created when a disturbance causes a wave to travel in a sine pattern
until it hits another wave or the boundary of the medium in which it is travelling
• We also studied the interference that occurs when two waves move in the same direction
• This is called superposition and results in
• Constructive interference: when two waves travel “in-phase”, with the same
wavelength (or wavelengths that are whole integers apart), or
• Destructive interference: when two waves travel “out-of-phase”, with different
wavelengths (one wavelength is half a wavelength apart from the other)

3. WHAT IS A STANDING WAVE?
• Standing waves occur
• between two waves moving in the opposite direction
as each other with equal amplitude, wavelength, and
frequency
• ONLY when the medium is vibrated at the right
amount of frequency (which depends on each
situation)

4. MATHEMATIC DEFINITION OF
A STANDING WAVE
• The mathematical expression of a standing wave can be determined using wave functions
• Moving right (increasing x): DR = A + sin(kx-wt)
• Moving left (decreasing x) DL = A + sin(kx+wt)
• Using the law of superposition:
• DR + DL = [A + sin(kx-wt)] + [A + sin(kx+wt)]
= A [sin(kx-wt) - sin(kx+wt)]
• Using trig identity sin(a-b) + sin (a+b) = 2sin(a)cos(b), the following key equation can be
determined
Key Equation: D(x,t) = 2Asin(kx)cos(wt)

5. HOW DOES A STANDING
WAVES LOOK LIKE?
• The purple wave represents two waves moving in opposite directions which are overlapping
• NOTE: When the crests or valleys of the two waves cross over, maximum amplitude is achieved (A(x) = +/- 2A). These are
the anti-nodes.
• NOTE: When the waves are out of phase, the wave appears flat.The points where A(x) = 0 are called nodes.

6. NODES AND ANTINODES
Anti-node
Node
• As seen in this figure, the nodes appear half a wavelength ( ) away
from each other; this is also true for the anti-nodes
• In total, a node and anti-node are a quarter of a wavelength apart
from each other ( )
NodeNode
Anti-node
x
2A
-2A
A(x)

7. STANDING WAVES ON
STRINGS
• Now that we have learned the background to
standing waves, we can apply this to strings
• If a string has two fixed ends, then it has a length
• Each end has a node —> L = or
• Using and , we can rearrange these
equations to get the fundamental frequency :
L

8. FUNDAMENTAL HARMONICS
• The fundamental frequency, or fundamental harmonic,
represents the frequency of a standing wave when n=1
• This equation can be manipulated to determine fundamental
harmonic when n = 1, 2, 3, 4, etc.
• Therefore, the equation is
First Harmonic Second Harmonic Third Harmonic
n = 1 n = 2 n = 3

9. PRACTICE
• Sandy loves to play guitar and has recently learned about standing waves
in her Physics class. She decides to calculate the mass of the guitar string
using her knowledge of standing waves and information given in the
instruction manual for her guitar. She knows:
• The tension of the guitar string is 79N
• The length of the string is 45cm, and
• The frequency of the third harmonic of her guitar is 1.449 kHz
• What is the mass of the guitar string in grams?

10. HINTS
• The wavelength of a third harmonic is
• Remember,
• Watch your units!!

11. SOLUTION

12. THANKS FOR WATCHING!