Chapter 14, Problem 5P

(a)

To determine

The value of the sum of two wave functions $y_1+y_2$ at given points.

Answer to Problem 5P

The value of the sum of two wave functions $y_1+y_2$ at given points is $-1.647\text{cm}$.

Explanation of Solution

Given information:The first wave function is $3.0\cos \left(4.0x-1.6t\right)$, the second wave function is $4.0\sin \left(5.0x-2.0t\right)$, the position of the element is $1\text{cm}$ and time of the wave function is $1\text{s}$.

The sum of two wave functions is,

$y=y_1+y_2$

• $y$ is the sum of the two wave functions.
• $y_1$ is the first wave function.
• $y_2$ is second wave function.

Substitute $3.0\cos \left(4.0x-1.6t\right)$ for $y_1$ and $4.0\sin \left(5.0x-2.0t\right)$.

$y=3.0\cos \left(4.0x-1.6t\right)+4.0\sin \left(5.0x-2.0t\right)$

Substitute $1\text{cm}$ for $x$ and $1\text{s}$ for $t$ in above expression to find $y$.

$\begin{array}{c}y=3.0\cos \left(4.0\times 1\text{cm}-1.6\times 1\text{s}\right)+4.0\sin \left(5.0\times 1\text{cm}-2.0\times 1\text{s}\right)\\ =-1.647\text{cm}\end{array}$

Conclusion:

Therefore, the value of the sum of two wave functions $y_1+y_2$ at given points is $-1.647\text{cm}$.

(b)

To determine

The value of the sum of two wave functions $y_1+y_2$ at given points.

Answer to Problem 5P

The value of the sum of two wave functions $y_1+y_2$ at given points is $-6.022\text{cm}$.

Explanation of Solution

Given information:The first wave function is $3.0\cos \left(4.0x-1.6t\right)$, the second wave function is $4.0\sin \left(5.0x-2.0t\right)$, the position of the element is $1\text{cm}$ and time of the wave function is $0.500\text{s}$.

The sum of two wave functions is,

$y=y_1+y_2$

• $y$ is the sum of the two wave functions.
• $y_1$ is the first wave function.
• $y_2$ is second wave function.

Substitute $3.0\cos \left(4.0x-1.6t\right)$ for $y_1$ and $4.0\sin \left(5.0x-2.0t\right)$.

$y=3.0\cos \left(4.0x-1.6t\right)+4.0\sin \left(5.0x-2.0t\right)$

Substitute $1\text{cm}$ for $x$ and $0.500\text{s}$ for $t$ to find the $y$.

$\begin{array}{c}y=3.0\cos \left(4.0\times 1\text{cm}-1.6\times 0.500\text{s}\right)+4.0\sin \left(5.0\times 1\text{cm}-2.0\times 0.500\text{s}\right)\\ =-6.022\text{cm}\end{array}$

Conclusion:

Therefore, the value of the sum of two wave functions $y_1+y_2$ at given points is $-6.022\text{cm}$.

(c)

To determine

The value of the sum of two wave functions $y_1+y_2$ at given points.

Answer to Problem 5P

The value of the sum of two wave functions $y_1+y_2$ at given points is $1.15\text{cm}$.

Explanation of Solution

Given info:The first wave function is $3.0\cos \left(4.0x-1.6t\right)$, the second wave function is $4.0\sin \left(5.0x-2.0t\right)$, the position of the element is $0.500\text{cm}$ and time of the wave function is $0\text{s}$.

The sum of two wave functions is,

$y=y_1+y_2$

• $y$ is the sum of the two wave functions.
• $y_1$ is the first wave function.
• $y_2$ is second wave function.

Substitute $3.0\cos \left(4.0x-1.6t\right)$ for $y_1$ and $4.0\sin \left(5.0x-2.0t\right)$,

$y=3.0\cos \left(4.0x-1.6t\right)+4.0\sin \left(5.0x-2.0t\right)$

Substitute $0.500\text{cm}$ for $x$ and $0\text{s}$ for $t$ to find the $y$.

$\begin{array}{c}y=3.0\cos \left(4.0\times 0.500\text{cm}-1.6\times 0\text{s}\right)+4.0\sin \left(5.0\times 0.500\text{cm}-2.0\times 0\text{s}\right)\\ =1.15\text{cm}\end{array}$

Conclusion:

Therefore, the value of the sum of two wave functions $y_1+y_2$ at given points is $1.15\text{cm}$.