Standing Wave on a string

2. DEFINITION OF A STANDING WAVE
- A “standing wave” is essentially a wave that results from an interaction with the
exact same wave, however they are both travelling in opposite directions .
- A standing wave is only created when the frequency is optimal (also known as the
resonant frequencies of the object the wave is traveling through).
- When the frequency is correct the waves will superposition themselves to form a
special pattern shown below.

3. WHAT DOES THIS MEAN?
- In a standing wave you will observe that there are points which appear to be
“zero”.
- These “zero” or stationary points are also know as the nodes of a standing wave.
- The points of maximum displacement are know as the antinodes.
- These points are shown on the diagram below.

4. EQUATION OF A STANDING WAVE
We know that the two waves are coming towards one and other. Hence, we can set up
an equation for the standing wave:
D (x,t) = Asin(kx – wt) + Asin(kx + wt)
D (x,t) = A[ sin(kx – wt) + sin(kx + wt) ]
Using known trigonometric identifies this can be simplified down to:
D (x,t) = 2Asin(kx)cos(wt)
As seen above, the sin (red) part of the equation is clearly dependent on the position,
hence changes with x.
The cos (blue) part of the equation is dependent on the time, thus changes with t.
The 2A here represents the max amplitude of the wave.

5. MORE ON NODES AND ANTINODES
- Each node occurs at a fundamental frequency (m) at which the string resonates.
These are known as the “harmonics” of the string.
- These fundamental frequencies create different number of nodes as shown
below:
- At the first fundamental frequency of harmonic we see 2 nodes, and at the
second harmonic we three nodes and so on.

6. STANDING WAVE ON A GUITAR STRING
A standing wave can be created by plucking a guitar string to create a standing wave
with the amplitude of the standing wave being given by the equation:
A(x) = 0.3cm*sin(4π*x)
Questions:
1) Determine the amplitude and λ of the traveling wave function.
2) Determine the position of the first 2 nodes.
3) Determine the first harmonic frequency of the wave, if the wave speed is 395m/s
and the length of the guitar string is 50 centimeters.

7. SOLUTIONS TO QUESTION 1
Step 1: Compare the given wave form with that of a standing wave:
a: D (x,t) = 2Asin(kx)cos(wt) as we are only concerned with the position (x value) we
can ignore the cos part of the function for this question.
Here kx = 2πx / λ
b: A(x) = 0.3cm*sin(4π*x)
2A = 0.3cm Hence, A = 0.3 / 2 = 0.15
2πx / λ = 4πx Hence λ = 0.5

8. SOLUTION TO QUESTION 2
The nodes of a standing wave occur when the function is equal to zero.
Therefore:
A(x) = 0.3cm*sin(4π*x) = 0
So when sin(x) = 0, which occurs at n*π (where n is the position +/- 1)
Thus, when 4π*x = n*π
4x = n
x = n/4
1st node:
¼ cm
2nd node
½ cm

9. SOLUTION TO QUESTION 3
Recall:
V = f *λ
Given, v = 395 m/s
λ = 2L, thus 2*0.5m
f(harmonic 1) = v/λ
= 395/0.5
= 790Hz

10. REFERENCES
Unknown. "Standing Waves." Standing Waves. N.p., n.d. Web. 08 Mar. 2015.
Unknown. "Standing Wave Patterns." Standing Wave Patterns. N.p., n.d. Web. 08 Mar.
2015.