2. STANDING WAVES
β’ Are stationary (as opposed to travelling waves)
β’ Vs
STANDING WAVE TRAVELLING WAVE
3. STANDING WAVES
β’ Are the superposition of two
harmonic waves with equal
amplitude, frequency and
wavelengths but moving in
opposite direction
v
v
Resulting Standing Wave
from adding the two
harmonic waves
4. STANDING WAVES
β’ Can be generated by plucking a string with
both ends fixed
β’ Nodes are points with zero amplitudes
β’ Antinodes are points with maximum
amplitudes
5. STANDING WAVES ON STRINGS
β’ Strings with two fixed ends can only produce
standing waves with an integral number of half
wavelength called normal modes
β’ Ξ» π =
2πΏ
π
where L = string length
n = number of antinodes = 1, 2, 3, 4, β¦
β’ The fundamental frequency (1st harmonic) is the
lowest frequency (longest wavelength)
β’ π1 =
1
2πΏ
π
π
where T = tension in the string
π = linear mass density of the string =
πππ π ππ π π‘ππππ
πππππ‘β ππ π π‘ππππ
β’ The allowed frequencies are called harmonics
β’ ππ = nπ1 n = 1, 2, 3, 4, β¦
6. QUESTION PART 1
Tom wants to make a violin for his sister as a birthday present.
Violins usually make sound frequencies ranging from
200~3000Hz. He has a few 30 cm long strings with linear mass
densities:
A 2.8 Γ 10β4
kg/m
B 4.0 Γ 10β4
kg/m
C 0.62 g/m
Which string should he use to make the violin in order to get a
fundamental frequency of 700Hz if the tension in the string is
kept at 70 N?
7. Hints
β’ What variables are given in the question?
β’ The fundamental frequency (π1), tension (T), and string length (L)
8. Hints
β’ What variables are given in the question?
β’ The fundamental frequency (π1), tension (T), and string length (L)
β’ Which equation to use when solving for linear mass density?
β’ π1 =
1
2πΏ
π
π
where T = tension in the string
π = linear mass density of the string =
πππ π ππ π π‘ππππ
πππππ‘β ππ π π‘ππππ
9. Solution β Tom should use string B
π1 = 700 Hz T = 70 N L = 30 cm = 0.30 m
π1 =
1
2πΏ
π
π
Solve for π
π1 Γ 2πΏ =
π
π
(π1 Γ 2πΏ) 2
=
π
π
π =
π
( π1Γ2πΏ)
2 =
70 π
(700 π»π§Γ2Γ0.30π)
2 = 3.97Γ 10β4 kg/ m
β 4.0 Γ 10β4
kg/m
10. QUESTION PART 2
The violin string broke after a few weeks, but Tom
doesnβt have anymore of the same string. If he
uses a string with linear mass density of 4.7 Γ
10β4
kg/m, what should the tension be in the
string in order to produce the same sound
frequency (700 Hz)?
11. Hints
β’ What variables are given in the question?
β’ The fundamental frequency (π1), linear mass density (π),
and string length (L)
12. Hints
β’ What variables are given in the question?
β’ The fundamental frequency (π1), linear mass density (π),
and string length (L)
β’ Which equation to use when solving for tension?
β’ π1 =
1
2πΏ
π
π
where T = tension in the string
π = linear mass density of the string =
πππ π ππ π π‘ππππ
πππππ‘β ππ π π‘ππππ
13. Solution
π1 = 700 Hz π = 4.7 Γ 10β4
kg/m L = 30 cm = 0.30 m
π1 =
1
2πΏ
π
π
Solve for T
π1 Γ 2πΏ =
π
π
(π1 Γ 2πΏ) 2
=
π
π
T = (π1 Γ 2πΏ) 2
Γ π = (700π»π§ Γ 2 Γ 0.30π) 2
Γ 4.7 Γ 10β4
kg/m
= 82.9 N
β 83 N