Representing Waves as Sine and Cosine Functions - AP Physics 2

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Question

Which of the following electromagnetic waves has the highest frequency?

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Answer

The frequency of a wave is directly related to the energy in the wave. The most energetic waves in the electromagnetic spectrum are gamma rays. Gamma rays are typically released from quasars, ultra-dense dying stars, and from atomic bomb blasts, which gives you the scope of the amount of energy involved with gamma rays.

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Question

Equation of sound wave 1:

$Asin(\omega_1t+\phi_1)$

Equation of sound wave 2:

$Asin(\omega_2t+\phi_2)$

Determine at what time there will be complete constructive interference between the two if:

$\omega_1=.5$\frac{\pi}{s}$$

$\omega_2=1$\frac{\pi}{s}$$

$\phi_2=\pi$

$\phi_1=\frac{\pi}{2}$$

Tap to reveal answer

Answer

The will be destructive interference when

$Asin(\omega_1t+\phi_1)+Asin(\omega_2t+\phi_2)=0$

$Asin(\omega_1t+\phi_1)=-Asin(\omega_2t+\phi_2)$

$sin(\omega_1t+\phi_1)=-sin(\omega_2t+\phi_2)$

Using

$-sin(x)=sin(x+\pi)$

$sin(\omega_1t+\phi_1)=sin(\omega_2t+\phi_2+\pi)$

$\omega_1t+\phi_1=\omega_2t+\phi_2+\pi$

$\omega_1t-\omega_2t=\phi_2+\pi-\phi_1$

$(\omega_1-\omega_2)t=\phi_2+\pi-\phi_1$

$t=\frac{\phi_2+\pi-\phi_1}{\omega_1-\omega_2}$$

Plugging in values:

$t=\frac{\frac{\pi}{2}$+\pi-\pi}{$\frac{\pi}{s}$-$\frac{.5\pi}{s}$}$

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Question

For open pipes, the formula for wave patterns at any given time can be given by a Fourier Sine Series which is given as the infinite sum:

$y(x)=\sum_${n=0}^{\infty}$ sin($\frac{n\pi x}{l}$)$ when is an integer, and is the length of the pipe. Each individual value of is called a harmonic.

What is the angular frequency of the second harmonic?

Tap to reveal answer

Answer

Frequency, when given an equation in sine or cosine notation, is the value inside the parenthesis. For the second harmonic , the frequency is:

$\frac{2\pi }{l}$

← Didn't Know|Knew It →

Question

Which of the following electromagnetic waves has the highest frequency?

Tap to reveal answer

Answer

The frequency of a wave is directly related to the energy in the wave. The most energetic waves in the electromagnetic spectrum are gamma rays. Gamma rays are typically released from quasars, ultra-dense dying stars, and from atomic bomb blasts, which gives you the scope of the amount of energy involved with gamma rays.

← Didn't Know|Knew It →

Question

Equation of sound wave 1:

$Asin(\omega_1t+\phi_1)$

Equation of sound wave 2:

$Asin(\omega_2t+\phi_2)$

Determine at what time there will be complete constructive interference between the two if:

$\omega_1=.5$\frac{\pi}{s}$$

$\omega_2=1$\frac{\pi}{s}$$

$\phi_2=\pi$

$\phi_1=\frac{\pi}{2}$$

Tap to reveal answer

Answer

The will be destructive interference when

$Asin(\omega_1t+\phi_1)+Asin(\omega_2t+\phi_2)=0$

$Asin(\omega_1t+\phi_1)=-Asin(\omega_2t+\phi_2)$

$sin(\omega_1t+\phi_1)=-sin(\omega_2t+\phi_2)$

Using

$-sin(x)=sin(x+\pi)$

$sin(\omega_1t+\phi_1)=sin(\omega_2t+\phi_2+\pi)$

$\omega_1t+\phi_1=\omega_2t+\phi_2+\pi$

$\omega_1t-\omega_2t=\phi_2+\pi-\phi_1$

$(\omega_1-\omega_2)t=\phi_2+\pi-\phi_1$

$t=\frac{\phi_2+\pi-\phi_1}{\omega_1-\omega_2}$$

Plugging in values:

$t=\frac{\frac{\pi}{2}$+\pi-\pi}{$\frac{\pi}{s}$-$\frac{.5\pi}{s}$}$

← Didn't Know|Knew It →

Question

For open pipes, the formula for wave patterns at any given time can be given by a Fourier Sine Series which is given as the infinite sum:

$y(x)=\sum_${n=0}^{\infty}$ sin($\frac{n\pi x}{l}$)$ when is an integer, and is the length of the pipe. Each individual value of is called a harmonic.

What is the angular frequency of the second harmonic?

Tap to reveal answer

Answer

Frequency, when given an equation in sine or cosine notation, is the value inside the parenthesis. For the second harmonic , the frequency is:

$\frac{2\pi }{l}$

← Didn't Know|Knew It →

Question

Which of the following electromagnetic waves has the highest frequency?

Tap to reveal answer

Answer

The frequency of a wave is directly related to the energy in the wave. The most energetic waves in the electromagnetic spectrum are gamma rays. Gamma rays are typically released from quasars, ultra-dense dying stars, and from atomic bomb blasts, which gives you the scope of the amount of energy involved with gamma rays.

← Didn't Know|Knew It →

Question

Equation of sound wave 1:

$Asin(\omega_1t+\phi_1)$

Equation of sound wave 2:

$Asin(\omega_2t+\phi_2)$

Determine at what time there will be complete constructive interference between the two if:

$\omega_1=.5$\frac{\pi}{s}$$

$\omega_2=1$\frac{\pi}{s}$$

$\phi_2=\pi$

$\phi_1=\frac{\pi}{2}$$

Tap to reveal answer

Answer

The will be destructive interference when

$Asin(\omega_1t+\phi_1)+Asin(\omega_2t+\phi_2)=0$

$Asin(\omega_1t+\phi_1)=-Asin(\omega_2t+\phi_2)$

$sin(\omega_1t+\phi_1)=-sin(\omega_2t+\phi_2)$

Using

$-sin(x)=sin(x+\pi)$

$sin(\omega_1t+\phi_1)=sin(\omega_2t+\phi_2+\pi)$

$\omega_1t+\phi_1=\omega_2t+\phi_2+\pi$

$\omega_1t-\omega_2t=\phi_2+\pi-\phi_1$

$(\omega_1-\omega_2)t=\phi_2+\pi-\phi_1$

$t=\frac{\phi_2+\pi-\phi_1}{\omega_1-\omega_2}$$

Plugging in values:

$t=\frac{\frac{\pi}{2}$+\pi-\pi}{$\frac{\pi}{s}$-$\frac{.5\pi}{s}$}$

← Didn't Know|Knew It →

Question

For open pipes, the formula for wave patterns at any given time can be given by a Fourier Sine Series which is given as the infinite sum:

$y(x)=\sum_${n=0}^{\infty}$ sin($\frac{n\pi x}{l}$)$ when is an integer, and is the length of the pipe. Each individual value of is called a harmonic.

What is the angular frequency of the second harmonic?

Tap to reveal answer

Answer

Frequency, when given an equation in sine or cosine notation, is the value inside the parenthesis. For the second harmonic , the frequency is:

$\frac{2\pi }{l}$

← Didn't Know|Knew It →

Question

Which of the following electromagnetic waves has the highest frequency?

Tap to reveal answer

Answer

The frequency of a wave is directly related to the energy in the wave. The most energetic waves in the electromagnetic spectrum are gamma rays. Gamma rays are typically released from quasars, ultra-dense dying stars, and from atomic bomb blasts, which gives you the scope of the amount of energy involved with gamma rays.

← Didn't Know|Knew It →

Question

Equation of sound wave 1:

$Asin(\omega_1t+\phi_1)$

Equation of sound wave 2:

$Asin(\omega_2t+\phi_2)$

Determine at what time there will be complete constructive interference between the two if:

$\omega_1=.5$\frac{\pi}{s}$$

$\omega_2=1$\frac{\pi}{s}$$

$\phi_2=\pi$

$\phi_1=\frac{\pi}{2}$$

Tap to reveal answer

Answer

The will be destructive interference when

$Asin(\omega_1t+\phi_1)+Asin(\omega_2t+\phi_2)=0$

$Asin(\omega_1t+\phi_1)=-Asin(\omega_2t+\phi_2)$

$sin(\omega_1t+\phi_1)=-sin(\omega_2t+\phi_2)$

Using

$-sin(x)=sin(x+\pi)$

$sin(\omega_1t+\phi_1)=sin(\omega_2t+\phi_2+\pi)$

$\omega_1t+\phi_1=\omega_2t+\phi_2+\pi$

$\omega_1t-\omega_2t=\phi_2+\pi-\phi_1$

$(\omega_1-\omega_2)t=\phi_2+\pi-\phi_1$

$t=\frac{\phi_2+\pi-\phi_1}{\omega_1-\omega_2}$$

Plugging in values:

$t=\frac{\frac{\pi}{2}$+\pi-\pi}{$\frac{\pi}{s}$-$\frac{.5\pi}{s}$}$

← Didn't Know|Knew It →

Question

For open pipes, the formula for wave patterns at any given time can be given by a Fourier Sine Series which is given as the infinite sum:

$y(x)=\sum_${n=0}^{\infty}$ sin($\frac{n\pi x}{l}$)$ when is an integer, and is the length of the pipe. Each individual value of is called a harmonic.

What is the angular frequency of the second harmonic?

Tap to reveal answer

Answer

Frequency, when given an equation in sine or cosine notation, is the value inside the parenthesis. For the second harmonic , the frequency is:

$\frac{2\pi }{l}$

← Didn't Know|Knew It →

Knew It

Representing Waves as Sine and Cosine Functions - AP Physics 2

Which of the following electromagnetic waves has the highest frequency?

Equation of sound wave 1: Equation of sound wave 2: Determine at what time there will be complete constructive interference between the two if:

The will be destructive interference when Using Plugging in values:

For open pipes, the formula for wave patterns at any given time can be given by a Fourier Sine Series which is given as the infinite sum: when is an integer, and is the length of the pipe. Each individual value of is called a harmonic. What is the angular frequency of the second harmonic?

Frequency, when given an equation in sine or cosine notation, is the value inside the parenthesis. For the second harmonic , the frequency is:

Which of the following electromagnetic waves has the highest frequency?

Equation of sound wave 1: Equation of sound wave 2: Determine at what time there will be complete constructive interference between the two if:

The will be destructive interference when Using Plugging in values:

For open pipes, the formula for wave patterns at any given time can be given by a Fourier Sine Series which is given as the infinite sum: when is an integer, and is the length of the pipe. Each individual value of is called a harmonic. What is the angular frequency of the second harmonic?

Frequency, when given an equation in sine or cosine notation, is the value inside the parenthesis. For the second harmonic , the frequency is:

Which of the following electromagnetic waves has the highest frequency?

Equation of sound wave 1: Equation of sound wave 2: Determine at what time there will be complete constructive interference between the two if:

The will be destructive interference when Using Plugging in values:

For open pipes, the formula for wave patterns at any given time can be given by a Fourier Sine Series which is given as the infinite sum: when is an integer, and is the length of the pipe. Each individual value of is called a harmonic. What is the angular frequency of the second harmonic?

Frequency, when given an equation in sine or cosine notation, is the value inside the parenthesis. For the second harmonic , the frequency is:

Which of the following electromagnetic waves has the highest frequency?

Equation of sound wave 1: Equation of sound wave 2: Determine at what time there will be complete constructive interference between the two if:

The will be destructive interference when Using Plugging in values:

For open pipes, the formula for wave patterns at any given time can be given by a Fourier Sine Series which is given as the infinite sum: when is an integer, and is the length of the pipe. Each individual value of is called a harmonic. What is the angular frequency of the second harmonic?

Frequency, when given an equation in sine or cosine notation, is the value inside the parenthesis. For the second harmonic , the frequency is:

Equation of sound wave 1:

$Asin(\omega_1t+\phi_1)$

Equation of sound wave 2:

$Asin(\omega_2t+\phi_2)$

Determine at what time there will be complete constructive interference between the two if:

$\omega_1=.5$\frac{\pi}{s}$$

$\omega_2=1$\frac{\pi}{s}$$

$\phi_2=\pi$

$\phi_1=\frac{\pi}{2}$$

Frequency, when given an equation in sine or cosine notation, is the value inside the parenthesis. For the second harmonic , the frequency is:

$\frac{2\pi }{l}$

Equation of sound wave 1:

$Asin(\omega_1t+\phi_1)$

Equation of sound wave 2:

$Asin(\omega_2t+\phi_2)$

Determine at what time there will be complete constructive interference between the two if:

$\omega_1=.5$\frac{\pi}{s}$$

$\omega_2=1$\frac{\pi}{s}$$

$\phi_2=\pi$

$\phi_1=\frac{\pi}{2}$$

Frequency, when given an equation in sine or cosine notation, is the value inside the parenthesis. For the second harmonic , the frequency is:

$\frac{2\pi }{l}$

Equation of sound wave 1:

$Asin(\omega_1t+\phi_1)$

Equation of sound wave 2:

$Asin(\omega_2t+\phi_2)$

Determine at what time there will be complete constructive interference between the two if:

$\omega_1=.5$\frac{\pi}{s}$$

$\omega_2=1$\frac{\pi}{s}$$

$\phi_2=\pi$

$\phi_1=\frac{\pi}{2}$$

Frequency, when given an equation in sine or cosine notation, is the value inside the parenthesis. For the second harmonic , the frequency is:

$\frac{2\pi }{l}$

Equation of sound wave 1:

$Asin(\omega_1t+\phi_1)$

Equation of sound wave 2:

$Asin(\omega_2t+\phi_2)$

Determine at what time there will be complete constructive interference between the two if:

$\omega_1=.5$\frac{\pi}{s}$$

$\omega_2=1$\frac{\pi}{s}$$

$\phi_2=\pi$

$\phi_1=\frac{\pi}{2}$$

Frequency, when given an equation in sine or cosine notation, is the value inside the parenthesis. For the second harmonic , the frequency is:

$\frac{2\pi }{l}$