Physics, 06.01.2021 09:20 miyahvictoria
Two traveling sinusoidal waves are described by the wave functions y1 = 4.85 sin [(4.35x − 1270t)] y2 = 4.85 sin [(4.35x − 1270t − 0.250)] where x, y1, and y2 are in meters and t is in seconds. (a) What is the amplitude of the resultant wave function y1 + y2?
Answers: 2
.![y_1 = 4.85\, \sin[(4.35\, x - 1270\, t) + 0]](/tpl/images/1015/9293/3c764.png)
.![y_2 = 4.85\, \sin[(4.35\, x - 1270\, t) + (-0.250)]](/tpl/images/1015/9293/75412.png)
.
and 
share the same frequency and wavelength. The only distinction between these two waves is the 
in 
. 
would still be a sine wave. The amplitude of 
could be found without using calculus.
.
to 
and 
. Apply the sum-of-angle identity of sine to rewrite ![\begin{aligned}y_2 &= 4.85\, \sin[(\underbrace{4.35\, x - 1270\, t}_{a}) + (\underbrace{-0.250}_{b})]\\ &= 4.85 \, [\sin(4.35\, x - 1270\, t)\cdot \cos(-0.250) \\ &\quad\quad\quad\; + \cos(4.35\, x - 1270\, t)\cdot \sin(-0.250)] \end{aligned}](/tpl/images/1015/9293/2cf28.png)
.![\begin{aligned}& y_1 + y_2\\[0.5em] &= 4.85\, [\sin(4.35\, x - 1270\, t) \\ &\quad \quad \quad\;+\sin(4.35\, x - 1270\, t)\cdot \cos(-0.250) \\ &\quad\quad\quad\; + \cos(4.35\, x - 1270\, t)\cdot \sin(-0.250)] \\[0.5em] &= 4.85\, [\sin(4.35\, x - 1270\, t)\cdot (1 + \cos(-0.250)) \\ &\quad\quad\quad\; + \cos(4.35\, x - 1270\, t)\cdot \sin(-0.250)] \end{aligned}](/tpl/images/1015/9293/d46ed.png)
.
and 
that satisfy the following hypothetical equation?![\begin{aligned}& (4.85\, m)\cdot \sin((4.35\, x - 1270\, t) + c)\\&= 4.85\, [\sin(4.35\, x - 1270\, t)\cdot (1 + \cos(-0.250)) \\ &\quad\quad\quad\; + \cos(4.35\, x - 1270\, t)\cdot \sin(-0.250)] \end{aligned}](/tpl/images/1015/9293/87879.png)
.
.![\begin{aligned}& m\cdot \sin((4.35\, x - 1270\, t) + c)\\[0.5em]&=m\, \sin(4.35\, x - 1270\, t)\cdot \cos(c) \\ &\quad\quad + m\, \cos(4.35\, x - 1270\, t)\cdot \sin(c) \\[0.5em] &=\sin(4.35\, x - 1270\, t)\cdot (m\, \cos(c)) \\ &\quad\quad + \cos(4.35\, x - 1270\, t)\cdot (m\, \sin(c)) \end{aligned}](/tpl/images/1015/9293/f8914.png)
.
and 
, the following should be satisfied:
, and
.
:
.
.
.
.
.
.
.
back to the system to find 
isn't required for finding the amplitude of 
. 
radians.)
is a real number, the amplitude of 
.
.
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