Answer:
two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is yR (x, t) = 0.70 m sin⎛ ⎝3.00 m−1 x − 6.28 s−1 t + π/16 rad⎞ ⎠ . What are the angular frequency, wave number, amplitude, and phase shift of the individual waves?
ω
=
6.28
s
−
1
,
k
=
3.00
m−
1
,
φ
=
π
rad,
A
R
=
2
A
cos
(φ
2
)
,
A
=
0.37
m
Explanation:
y1
(
x
,
t
)
=
A
sin(
k
x
−
ω
t
+φ
)
,
y
2
(
x
,
t
)
=
A
sin
(
k
x
−
ω
t
)
.
from the principle of superposition which states that when two or more waves combine, there resultant wave is the algebriac sum of the individual waves
y1
(
x
,
t
)
=
A
sin(
k
x
−
ω
t
+φ
)
, is generaL form of thw wave eqaution
A=amplitude
k=angular wave number
ω=angular frequency
φ
=phase constant
k=2π/lambda
ω=2π/T
yR (x, t) = 0.70 m sin{3.00 m−1 x − 6.28 s−1 t + π/16 rad}....................*
two waves superposed to give the above, assuming they are moving in the +x direction
y1
(
x
,
t
)
=
A
sin(
k
x
−
ω
t
+φ
)
,
.....................1
y
2
(
x
,
t
)
=
A
sin
(
k
x
−
ω
t
)
...........................2
adding the two equation will give
A
sin(
k
x
−
ω
t
+φ
)+A
sin
(
k
x
−
ω
t
)
.................3
A(
sin(
k
x
−
ω
t
+φ
)+
sin
(
k
x
−
ω
t
)
),......................4
similar to the following trigonometry identity
sina+sinb=2cos(a-b)/2sin(a+b)/2
let a=
(
k
x
−
ω
t
b=k
x
−
ω
t
+φ
)
y(x,t)=2Acos(φ/2)sin(k
x
−
ω
t
+φ/2)
k=3m^-1
lambda=2π/k=2.09m
ω=6.28= T=2π/6.28
T=1s
φ/2=π/16
φ=π/8rad
amplitude
2Acos(φ/2)=0.70 m
A=0.7/2cos(π/8)
A=0.37
m
