A sinusoidal wave is special type of periodic function. It has continuous negative and positive half cycles. In every positive half-cycle, it starts from zero and reaches its positive maximum value then reduces again to zero. Similarly, in every negative half-cycle, it starts from zero and reaches its negative maximum value then reduces again to zero.
Formation of Sinusoidal Wave
If we rotate a straight line of a specific length in respect of its one end, the rotating end of the line reaches a certain height at every instant of the rotation. If we plot the location of the tip of the rotating straight line in respect of every instant of the rotation, we will get a certain pattern. The pattern represents the waveform of a sinusoidal wave. The amplitude of the waveform is equal to the length of the rotating radius (straight line). The time period of the sinusoidal wave is the time which the radius takes to complete one full rotation. The full rotation means 360° rotation. The wavelength of the sinusoidal waveform is the advancement of the wave during one cycle.
The frequency of this wave is nothing but the number of cycles created during one second. In other words, the number of cycles advances during one second.
Equation of Sinusoidal Wave
We can represent a sinusoidal wave with the following function.
Where Vm is the amplitude of the waveform. In other words, Vm is the maximum value to which the wave reaches during each half cycle.
Angular Frequency of Sinusoidal Wave
ω is the angular velocity of the rotating straight line. That means ω is the angle covered by the rotating straight line for one second. This is the reason, we refer to this ω as the angular velocity of the wave. We also refer to this ω as the angular frequency of the sinusoidal wave.
Frequency of Sinusoidal Wave
The rotating straight line covers the 2π radian angle for one revolution. Again the rotating straight line covers the ω radian angle for one second. Hence the rotating straight line rotates ω by 2π complete revolutions in one second. Therefore the rotating straight line creates ω by 2π complete cycles in one second. Therefore, we can express the frequency of the sinusoidal waveform as
Now the time taken by the wave to propagate one cycle is
This time is defined as the time period of the wave.
The time period is the time taken by a wave to advance one cycle.
Theta is an angle by which the wave is advanced at its origin. In many cases, there may be a certain magnitude at the origin that is when time t is zero. The magnitude of the waveform at t = 0 is,
Other Forms of Sinusoidal Wave
Generally, there are two forms of sinusoidal waves. These are the sine wave and the cosine wave function. The similar sine and cosine waves have the same pattern of the waveform. The only difference is when the sine wave reaches its maximum value the cosine wave reaches zero. The Laplace transform of sine and cosine waves are as follows.