time in Sine Wave equation

Question

According to Wikipedia,

The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation.

The formula for the Sine wave is,

A = Amplitude of the Wave ω = the angular frequency, specifies how many oscillations occur in a unit time interval, in radians per second φ, the phase, t = ?

Here ω, is the angular frequency i.e ,

It defines how many cycles of the oscillations are there.

Now let see the frequency,

Frequency is the number of occurrences of a repeating event per unit time

For example, if 100 events occur within 15 seconds the frequency is:

Events = 100;
Time = 15;
FREQ = Events/Time;

Which means there is total 100 cycles of wave in 15 seconds , am i correct ?

Events = 71;
Time = 15;
FREQ = Events/Time;
W = 2*PI*FREQ;   % 2*PI for circular path
A = 2 ;

Now what is the use of small 't' ? in the Sine wave equation , what time is it ?

Answer 1

In the context that you described, $t$ is a variable indicating time. That is, the wave will take on a value of $y(t)$ at the time instant $t$. If this is measured in seconds, then $\omega$ specifies the number of radians that the wave passes through in one second. And, as you noted, if the wave has a frequency of $\frac{100}{15}\ \text{Hz}$, then it will pass through 100 periods in 15 seconds.

Answer 2

Well to clear your doubt ,i would like you to ask you a question. What would be the value of angular frequency × Time period = ??

Okay,so in eq - $\omega \times t$ . if we put $T$ in place place of $t$, given that the wave is sinusoidal, the value to that eq would be $2\pi$. By definition $T$ is the time taken to complete one oscillation so when we put $T$ in place of $t$ , the value we get from the equation is - $2\pi$(as total distance traveled by a waves particle of a sinusoidal wave in time $t$=$T$ = $2\pi$). So we can say that significance of $t$ in the eq is that it tells the position of the wave's particle at time $t$ (note that I stated position of waves particle and not the wave itself because it that case we have another eq that is $\lambda = vT$ )

This equation works just like Distance =speed × time .

Answer 3

whenever you see the equation

Y= A Sin(wt+ @), the (wt+ @) section indicates a degree or radian value.With that in mind take Wt.

-From what we know W is angular frequency.That means Period or Cycle per second. in our case if we opt to take it in Radians ,then its 2π/time. If we take it in degrees then it becomes 360degrees/time. Its also revolutions per second.

-Again from what we know @ is an angle either in radians or Degrees

therefore if we take wt which is 2π/time,360degrees/second e.t.c and multiply it by (t).We are left with an angular value.. therefore wt is an angle that can be added to @ to get a phase value

think of it as Y= A Sin ( [22π/time] x t + 4π ) equals to Y= A Sin ( 22π + 4π )

or in degrees .

Y= A Sin ( [140degrees/time] x t + 40degrees ) ,Y= A Sin ( 140degrees + 40degrees )