Question

MESSAGE MY INSTRUCTOR FULL SCREEN PRINTER VERSION BACK NEXT Chapter 16, Problem 037 GO These two waves travel along the same string: y-(3.82 mm) sin(1.96x- 430rt) Y2 (5.64 mm) sin(1.96x-430xt + 0.780arad). What are (a) the amplitude and (b) the phase angle (relative to wave 1) of the resultant wave? (c) Ifa third wave of amplitude 4.54 mm is also to be sent along the string in the same direction as the first two waves, what should be its phase angle in order to maximize the amplitude of the new resultant wave? (a) Number 9.46 Units mm (b) Number Units 0.780 (c) Number 14 Units mm

Answer 1

given

y₁ = ( 3.82 mm ) sin( 1.96$\pi$x - 430$\pi$t )

y₁ = ( 5.64 mm ) sin( 1.96$\pi$x - 430$\pi$t + 0.78 $\pi$ rad )

y_m1 = 3.82 mm

y_m2 = 5.64 mm

$\phi$₁ = 0 rad

$\phi$₂ = 0.78 $\pi$ rad

according to sine and cosine rule

Y_m² = y_m1² + y_m2² - 2 y_m1 y_m2 cos( $\pi$ - $\phi$ ₂ )

and

sin ( $\pi$ - $\phi$ ) / Y_m = sin ( $\phi$ _R ) / y_m2

( a )  

Y_m² = 3.82² + 5.64² - 2 x 3.82 x 5.64 x cos( $\pi$ - 0.78$\pi$ )

Y_m² = 3.315

Y_m = 1.82 mm

( b )

sin ( $\pi$ - 0.78$\pi$ ) / 1.82 = sin ( $\phi$ _R ) / 5.64

sin ( $\pi$ - 0.78$\pi$ ) / 1.82 = sin ( $\phi$ _R ) / 5.64

sin ( $\phi$ _R ) = 0.01206 x 5.64 / 1.82  

$\phi$_R = 2.141 rad or 122.67 degrees

( c )

the waves are moving in the same direction

so the angle is same as above

$\phi$_R = 2.141 rad or 122.67 degrees