The equation of a standing wave is obtained by adding the displacements of two waves traveling in opposite directions (see figure). Assume that each of the waves has amplitude A, period T, and wavelength 2. The models for two such waves are Y₁ = A cos(2m (+-)) and y2 = A cos(2π(++)). Show that Y₁ + y2 = 2A cos(2) cos(2X). Y₁1 + Y2 = A cos(2π(⇒ − \)) + [ = A[cos(2m) cos(274) + sin(2Ç) sin(2π4)] + A[cos(2π) cos(24) - [ = 2A cos(2πt) cos(2x). t=0 yı V1+Y2] [Vi+Y2 [Ji+Y2 Y2
The equation of a standing wave is obtained by adding the displacements of two waves traveling in opposite directions (see figure). Assume that each of the waves has amplitude A, period T, and wavelength 2. The models for two such waves are Y₁ = A cos(2m (+-)) and y2 = A cos(2π(++)). Show that Y₁ + y2 = 2A cos(2) cos(2X). Y₁1 + Y2 = A cos(2π(⇒ − \)) + [ = A[cos(2m) cos(274) + sin(2Ç) sin(2π4)] + A[cos(2π) cos(24) - [ = 2A cos(2πt) cos(2x). t=0 yı V1+Y2] [Vi+Y2 [Ji+Y2 Y2
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.3: Trigonometric Functions Of Real Numbers
Problem 40E
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![The equation of a standing wave is obtained by adding the displacements of two waves traveling in opposite directions (see figure). Assume that each of the waves has amplitude A,
period T, and wavelength 2. The models for two such waves are
Y₁ = A cos(2π(-)) and y2 = A cos(2π(+)).
Show that
Y₁ + y2 = 2A cos(2πt) cos(2πX).
Y₁ + y2 = A cos(2π( - )) +
= A[cos(2ý) cos(2π) + sin(2Ç) sin(2Ã¥)] + A[cos(2ý) cos(2¬¥) –
= 2A cos(2πt) cos(2x).
t=0
t = {T
t=²T|
y1
y1
|Y1+J2
y1+y2
y₁+y2
Y2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2367e66-1ae4-4371-aa7b-db391deab840%2F7987e87b-e8ac-417d-a43b-49803b72afda%2Faprf04_processed.png&w=3840&q=75)
Transcribed Image Text:The equation of a standing wave is obtained by adding the displacements of two waves traveling in opposite directions (see figure). Assume that each of the waves has amplitude A,
period T, and wavelength 2. The models for two such waves are
Y₁ = A cos(2π(-)) and y2 = A cos(2π(+)).
Show that
Y₁ + y2 = 2A cos(2πt) cos(2πX).
Y₁ + y2 = A cos(2π( - )) +
= A[cos(2ý) cos(2π) + sin(2Ç) sin(2Ã¥)] + A[cos(2ý) cos(2¬¥) –
= 2A cos(2πt) cos(2x).
t=0
t = {T
t=²T|
y1
y1
|Y1+J2
y1+y2
y₁+y2
Y2
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