wave · verb I or T /weɪv/Full view
wave
B1 to raise your hand and move it from side to side in order to attract someone's attention or to say goodbye
Dictionary examples:

I waved to/at him, and he waved back.

She was so mad she wouldn't even wave goodbye to us.

She waves her hands around a lot when she's talking.

Learner example:

She waved at me and I smiled. (Preliminary English Test; B1; Chinese)

wave (MOVE SIDE TO SIDE)
C1 to move from side to side in the air or make something move this way
Dictionary examples:

The long grass waved in the breeze.

He started waving his arms around wildly.

Learner example:

I can picture myself standing in front of the "Casa Prosada," waving my arms and cheering with all the other patriots. (Certificate in Advanced English; C1; Spanish)

wave · noun C /weɪv/
wave (WATER)
B1 a line of higher water that moves across the surface of an area of water, especially the ocean
Dictionary example:

At night, I listened to the sound of the waves breaking/crashing against the shore.

Learner example:

I already know you surf and the waves at Reserva will be huge this weekend [so] we will surely surf together. (Preliminary English Test; B1; Portuguese)

wave (ENERGY)
B2 the pattern in which some types of energy, such as sound, light and heat, are spread or carried
Dictionary example:

radio waves

Learner example:

Indeed, it is thought that using a mobile phone may cause brain damage owing to magnetic waves. (First Certificate in English; B2; French)

wave (GROUP)
C2 a group of people or things that arrive or happen together or in a short period of time
Dictionary examples:

There has been a wave of robberies in the region.

Another wave of people flowed into the theater.

Learner example:

These people, who belong mostly to the middle class, and some other[s] like explorers [...] or youngsters seeking some adventure, produce the big wave of tourists to different countries. (Certificate of Proficiency in English; C2; Greek)

a wave of hatred/enthusiasm/sadness, etc.
C2 when you suddenly feel an emotion
Dictionary example:

She felt a sudden wave of sadness.

Learner example:

As he counted the money, a wave of shame and horror hit Berthold. (Certificate of Proficiency in English; C2; Portuguese)

make waves
C2 to cause problems by asking difficult questions or criticizing
Dictionary example:

He has earned a reputation for making waves during political campaigns.

Learner example:

If we don't want to regret anything, it is time we started to make waves by s[c]rutinizing our history. (Certificate of Proficiency in English; C2; Turkish)

wave (HAND)
C2 when you raise your hand and move it from side to side in order to attract someone's attention or to say hello or goodbye
Dictionary example:

She gave a little wave as the train left.

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