Question

wave properties review
directions: answer the question about frequency.

what is the frequency of the wave above?
____ hz

Explanation:

Step1: Determine Period (T)

The period \( T \) is the time for one full wave cycle. From the graph, one cycle (peak to peak or trough to trough) occurs every \( 2 \) seconds? Wait, no—let's count the cycles. From \( t=0 \) to \( t=8 \) seconds, how many cycles? Let's see: at \( t=0 \), starts, then at \( t=2 \), one cycle? Wait, no, let's look at the wave. The wave has peaks and troughs. Let's count the number of cycles in 8 seconds. Wait, from \( t=0 \) to \( t=8 \), how many full cycles? Let's see the graph: at \( t=0 \), it's at 0, then goes up, down, and at \( t=2 \), back to 0? Wait, no, maybe the period is 2 seconds? Wait, no, let's count the number of cycles. Wait, the graph shows that in 8 seconds, how many cycles? Let's see: from \( t=0 \) to \( t=8 \), let's count the number of times it completes a cycle. Let's see the wave: at \( t=0 \), starts, then at \( t=2 \), one cycle? Wait, no, maybe the period \( T \) is 2 seconds? Wait, no, let's check the number of cycles. Wait, the wave has 4 cycles in 8 seconds? Wait, no, let's look again. Wait, the x-axis is time (s), from 0 to 8. Let's count the number of full waves (cycles) in 8 seconds. Let's see: at \( t=0 \), it's at 0, then goes up, down, and at \( t=2 \), back to 0? Wait, no, maybe each cycle is 2 seconds? Wait, no, let's count the number of cycles. Wait, the graph has peaks and troughs. Let's see: from \( t=0 \) to \( t=2 \): one cycle? Then \( t=2 \) to \( t=4 \): second cycle? \( t=4 \) to \( t=6 \): third cycle? \( t=6 \) to \( t=8 \): fourth cycle? Wait, so in 8 seconds, there are 4 cycles? Wait, no, that can't be. Wait, maybe I'm miscounting. Wait, the period is the time for one cycle. Let's find the time between two consecutive peaks (or troughs). Let's see the first peak is at \( t=0.5 \)? No, the graph is a wave with time on x-axis. Wait, maybe the period \( T \) is 2 seconds? Wait, no, let's calculate frequency \( f = \frac{1}{T} \), and also \( f = \frac{\text{number of cycles}}{\text{time}} \). Let's count the number of cycles in 8 seconds. Looking at the graph, from \( t=0 \) to \( t=8 \), how many full cycles? Let's see: at \( t=0 \), it's at 0, then goes up, down, and at \( t=2 \), back to 0 (completing one cycle). Then at \( t=4 \), another cycle, \( t=6 \), third, \( t=8 \), fourth. So in 8 seconds, 4 cycles. Therefore, number of cycles \( n = 4 \), time \( t = 8 \) s. Then frequency \( f = \frac{n}{t} = \frac{4}{8} = 0.5 \) Hz? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, maybe the period is 2 seconds? Wait, no, let's check again. Wait, the wave: at \( t=0 \), starts at 0, goes up to a peak, down to a trough, and back to 0 at \( t=2 \). So that's one cycle (period \( T = 2 \) s). Then in 8 seconds, how many cycles? \( 8 / 2 = 4 \) cycles. So frequency \( f = \frac{1}{T} = \frac{1}{2} = 0.5 \) Hz? Wait, no, that would be if period is 2. Wait, but maybe the period is 2 seconds? Wait, no, let's see: the time between two consecutive zero crossings (going from 0 up, then down to 0) is 2 seconds? Wait, no, from \( t=0 \) to \( t=2 \), it goes up, down, and back to 0. So that's one full cycle (peak to peak or trough to trough). So period \( T = 2 \) s. Then frequency \( f = \frac{1}{T} = 0.5 \) Hz? Wait, but maybe I'm wrong. Wait, let's count the number of cycles in 8 seconds. If in 8 seconds, there are 4 cycles, then \( f = 4 / 8 = 0.5 \) Hz. Yes, that makes sense. So the frequency is 0.5 Hz? Wait, no, wait, maybe the period is 2 seconds, so frequency is 1/2 = 0.5 Hz. Wait, but let's check again. The graph: at \( t=0 \), 0, then at \( t=2 \), back to 0 (after a peak and trough). So that's one cycle (period 2 s). Then in 8 s, 4 cycles. So frequency \( f = 4 / 8 = 0.5 \) Hz. Wait, but maybe the period is 2 seconds, so frequency is 0.5 Hz. Alternatively, maybe I miscounted the cycles. Wait, let's look at the graph again. The wave has 4 cycles in 8 seconds, so frequency is 4/8 = 0.5 Hz. So the frequency is 0.5 Hz? Wait, no, maybe the period is 2 seconds, so frequency is 0.5 Hz.

Step2: Calculate Frequency

Frequency \( f \) is the number of cycles per second. The formula is \( f = \frac{\text{Number of Cycles}}{\text{Time}} \). From the graph, in 8 seconds, there are 4 cycles (let's confirm: at \( t=0 \), start; \( t=2 \), 1 cycle; \( t=4 \), 2 cycles; \( t=6 \), 3 cycles; \( t=8 \), 4 cycles). So number of cycles \( n = 4 \), time \( t = 8 \) s. Then \( f = \frac{4}{8} = 0.5 \) Hz. Alternatively, period \( T \) is the time for one cycle, which is \( 8 \, \text{s} / 4 = 2 \, \text{s} \). Then \( f = \frac{1}{T} = \frac{1}{2} = 0.5 \, \text{Hz} \).

Answer:

0.5