Standing Waves

The objective of the experiment is to observe what are the conditions for created a resonant wave on a string.

Procedure:
See Physics 4C Lab Manual (Martin Sydney Mason), Experiment 4.

Data
Mass1=200g

Mass2= 50 g

Analysis

(1) v=sqrt(T/μ), T1=m1g=0.200 kg*9.81m/s^2, T2=m2g=0.050kg*9.81m/s^2, μ=0.00323kg/2.665m

(2) v=λf, λ=v/f; v1= 40.23 m/s, v2= 20.12 m/s

(3) n= 2L/λ

Trial 1

Harmonic	Wavelength (m)	n (Rounded to nearest whole number)
1	3.09	2
2	1.68	3
3	1.06	5
4	0.805	7
5	0.639	8
6	0.529	10
7	0.457	12
8	0.398	13
9	0.353	15
10	0.317	17

Trial 2

Harmonic	Wavelength (m)	n (Rounded to nearest whole number)
1	3.35	2
2	1.55	3
3	1.06	5
4	0.805	7
5	0.629	8
6	0.503	11

Question 4) v1/v2=40.23 m/s/20.12 m/s= 2.00
sqrt(T1/μ)/sqrt(T2/μ)=sqrt(4)=2 
Yes, the experimental ratio of wave speeds equals the theoretical value.

Question 5) Yes, the values of f for the nth harmonic are f=n*f1, assuming the frequencies generated by the mechanical vibrator are fn ± 0.5 Hz.

Question 6) The ratios of the frequencies between the trials are: f12/f22= 1.85, f13/f23= 2, f14/f24= 2, f15/f25= 1.97, and f16/f26= 1.9. The pattern is that the frequency of the harmonic of the first trial is double the corresponding frequency for the second trial. The is because frequency is proportional to velocity which is proportional to the square root of tension. 

Error Analysis
Assumed uncertainties:
mstring = 3.23 ± 0.01 g
Lstring= 2.665 ± 0.05 cm
m1= 200 g ± 1g
m2 = 50 g ± 1g   
Frequency of mechanical vibrator: f ± 0.5 Hz 

Since there are so many variables, conversions, and calculations, it is reasonable to see certain values, especially calculated values for n, be so off. 

Summary 
The conditions for resonance can be very specific, as there are many factors involved within a system. Observations of resonance can also be quite lengthy, as this experiment considered number of nodes, wavelength, frequency, and wave speed.