1. Sinusoid Seventeenth Meeting

2. Sine Wave: Amplitude The amplitude is the maximum displacement of the sine wave from its mean (average) position. Simulation

3. Sine Wave: Cycle, Frequency and Period Frequency (f) The number of cycles per second, Example: A sine wave with 5 cycles per second is said to have a frequency of 5 Hz (hertz) Cycle The basic shape of the waveform that repeats indefinitely. Period (T) The time taken to complete one cycle T = 1/f The mains electricity supply is sinusoidal, with a frequency of 50 Hz. What is T? 1/50 = 0.02 s One Cycle Simulation

4. Sine Wave: Phase Phase, or, more correctly, phase shift, Is how far a sine wave is shifted along the horizontal axis relative to another sine wave taken as a reference The blue sine wave is shifted 1/4 cycle to the right of the reference sine wave If a sine wave to be generated by the rotating line a, then a sine wave lagging by a quarter of a cycle is generated by a line b at 90 degrees to line a. (Ninety degrees = ¼ cycle) why? because a complete revolution, 360 degrees, corresponds to one complete cycle of a sine wave.) Simulation

5. Sine wave: Equation y = a sin(2πft + φ) y represents displacement at time t a represents the amplitude f is the frequency and φ is the phase The term (2πft + φ) represents an angle that is growing as time passes. This angle is measured in radians rather than degrees. For the following sine wave, it is clear that the amplitude a has the value 5 volts. The values f and φ are not so obvious. φ is a quarter of a cycle is 90 degrees, or π/2 radians. Since the sine wave lags behind the reference sine wave, so φ = – π/2 radians. (radian = 57.3 degrees) The equation for the sine wave is: y = 5 sin(200πt – π/2) volts

6. Periodic Waves y = 2 sin 2000 πt. y = 2 cos 2000 πt A cosine wave can be regarded as a sine wave shifted in phase by a quarter of a cycle sin wave cos wave

7. Sine & Cosine sine function v(t) = A sin (wt). A = amplitude w (omega) = angular frequency in rad/s. ƒ = frequency in Hz, T = the period in seconds of one cycle Cosine function v(t) = A sin (wt).

8. Square Wave The infinite extent of the spectrum results from the corners on the square waveform. These corners are assumed to be perfectly sharp The more we add, the sharper the corner With only the first five harmonic,

9. Some Fourier Series Partial sums of some Fourier series, up to and including ninth harmonic