What is the formula for a sinusoidal wave?
2 Answers
A sinusoidal wave can be described using a mathematical function that represents its amplitude, frequency, phase, and vertical shift. The general formula for a sinusoidal wave is:
\[ y(t) = A \sin(2 \pi f t + \phi) + C \]
where:
- \( y(t) \) is the value of the wave at time \( t \).
- \( A \) is the amplitude of the wave, which represents the peak value of the wave from its central position.
- \( \sin \) represents the sine function. Alternatively, you might see the cosine function, \(\cos\), used in similar contexts.
- \( 2 \pi f \) is the angular frequency, where \( f \) is the frequency of the wave in hertz (Hz), and \( 2 \pi \) converts the frequency to radians per second.
- \( t \) is the time variable.
- \( \phi \) is the phase shift, which indicates the horizontal shift of the wave. A positive \( \phi \) shifts the wave to the left, and a negative \( \phi \) shifts it to the right.
- \( C \) is the vertical shift, which moves the wave up or down relative to the horizontal axis.
### Detailed Explanation:
1. **Amplitude (\( A \))**: This is the maximum value the wave reaches from its central or equilibrium position. For instance, in a sound wave, this would correspond to the loudness.
2. **Frequency (\( f \))**: This is how many cycles of the wave occur in one second. Higher frequency means more cycles per second and thus a higher pitch in sound waves.
3. **Angular Frequency (\( \omega \))**: This is often used instead of \( 2 \pi f \). It is related to the frequency by the formula \( \omega = 2 \pi f \). It describes how quickly the wave oscillates in radians per second.
4. **Phase Shift (\( \phi \))**: This parameter determines where the wave starts in its cycle relative to a reference point. It can be used to align waves or to describe the timing of wave phenomena.
5. **Vertical Shift (\( C \))**: This adjusts the central position of the wave. If \( C \) is zero, the wave oscillates around the horizontal axis. If \( C \) is positive or negative, it shifts the wave up or down.
### Example
If you have a sinusoidal wave with an amplitude of 3, a frequency of 5 Hz, a phase shift of \( \pi/4 \) radians, and a vertical shift of -2, the equation would be:
\[ y(t) = 3 \sin(2 \pi \cdot 5 t + \pi/4) - 2 \]
This equation describes how the wave behaves over time. By adjusting \( A \), \( f \), \( \phi \), and \( C \), you can model different types of sinusoidal waves for various applications, including sound waves, alternating current (AC) in electrical circuits, and more.
\[ y(t) = A \sin(2 \pi f t + \phi) + C \]
where:
- \( y(t) \) is the value of the wave at time \( t \).
- \( A \) is the amplitude of the wave, which represents the peak value of the wave from its central position.
- \( \sin \) represents the sine function. Alternatively, you might see the cosine function, \(\cos\), used in similar contexts.
- \( 2 \pi f \) is the angular frequency, where \( f \) is the frequency of the wave in hertz (Hz), and \( 2 \pi \) converts the frequency to radians per second.
- \( t \) is the time variable.
- \( \phi \) is the phase shift, which indicates the horizontal shift of the wave. A positive \( \phi \) shifts the wave to the left, and a negative \( \phi \) shifts it to the right.
- \( C \) is the vertical shift, which moves the wave up or down relative to the horizontal axis.
### Detailed Explanation:
1. **Amplitude (\( A \))**: This is the maximum value the wave reaches from its central or equilibrium position. For instance, in a sound wave, this would correspond to the loudness.
2. **Frequency (\( f \))**: This is how many cycles of the wave occur in one second. Higher frequency means more cycles per second and thus a higher pitch in sound waves.
3. **Angular Frequency (\( \omega \))**: This is often used instead of \( 2 \pi f \). It is related to the frequency by the formula \( \omega = 2 \pi f \). It describes how quickly the wave oscillates in radians per second.
4. **Phase Shift (\( \phi \))**: This parameter determines where the wave starts in its cycle relative to a reference point. It can be used to align waves or to describe the timing of wave phenomena.
5. **Vertical Shift (\( C \))**: This adjusts the central position of the wave. If \( C \) is zero, the wave oscillates around the horizontal axis. If \( C \) is positive or negative, it shifts the wave up or down.
### Example
If you have a sinusoidal wave with an amplitude of 3, a frequency of 5 Hz, a phase shift of \( \pi/4 \) radians, and a vertical shift of -2, the equation would be:
\[ y(t) = 3 \sin(2 \pi \cdot 5 t + \pi/4) - 2 \]
This equation describes how the wave behaves over time. By adjusting \( A \), \( f \), \( \phi \), and \( C \), you can model different types of sinusoidal waves for various applications, including sound waves, alternating current (AC) in electrical circuits, and more.
The formula for a sinusoidal wave, which is a mathematical representation of a wave that oscillates with time, is given by:
\[ y(t) = A \sin(\omega t + \phi) \]
Here's a breakdown of each term in the formula:
1. **\( y(t) \)**: The displacement or value of the wave at time \( t \).
2. **\( A \)**: The amplitude of the wave, which represents the maximum value or height of the wave from its equilibrium position.
3. **\( \omega \)**: The angular frequency of the wave, which is related to the frequency \( f \) by the formula \( \omega = 2\pi f \). It represents how quickly the wave oscillates.
4. **\( t \)**: The time variable.
5. **\( \phi \)**: The phase angle, which represents the initial angle at \( t = 0 \). It determines the wave's horizontal shift.
Alternatively, the sinusoidal wave can also be represented using a cosine function:
\[ y(t) = A \cos(\omega t + \phi) \]
The sine and cosine functions are essentially the same waveform but shifted by a phase difference of \( \pi/2 \) radians (or 90 degrees).
\[ y(t) = A \sin(\omega t + \phi) \]
Here's a breakdown of each term in the formula:
1. **\( y(t) \)**: The displacement or value of the wave at time \( t \).
2. **\( A \)**: The amplitude of the wave, which represents the maximum value or height of the wave from its equilibrium position.
3. **\( \omega \)**: The angular frequency of the wave, which is related to the frequency \( f \) by the formula \( \omega = 2\pi f \). It represents how quickly the wave oscillates.
4. **\( t \)**: The time variable.
5. **\( \phi \)**: The phase angle, which represents the initial angle at \( t = 0 \). It determines the wave's horizontal shift.
Alternatively, the sinusoidal wave can also be represented using a cosine function:
\[ y(t) = A \cos(\omega t + \phi) \]
The sine and cosine functions are essentially the same waveform but shifted by a phase difference of \( \pi/2 \) radians (or 90 degrees).